Equivalent generating vectors of finitely generated modules over commutative rings
aa r X i v : . [ m a t h . A C ] D ec EQUIVALENT GENERATING VECTORS OF FINITELYGENERATED MODULES OVER COMMUTATIVE RINGS
LUC GUYOT
Abstract.
Let R be a commutative ring with identity and let M be an R -module which is generated by µ elements but not fewer. We denote bySL n ( R ) the group of the n × n matrices over R with determinant . Wedenote by E n ( R ) the subgroup of SL n ( R ) generated by the the matriceswhich differ from the identity by a single off-diagonal coefficient. Given n ≥ µ and G ∈ { SL n ( R ) , E n ( R ) } , we study the action of G by matrixright-multiplication on V n ( M ) , the set of elements of M n whose compo-nents generate M . Assuming that M is finitely presented and that R is anelementary divisor ring or an almost local-global coherent Prüfer ring, weobtain a description of V n ( M ) /G which extends the author’s earlier resulton finitely generated modules over quasi-Euclidean rings [Guy17]. Introduction
Rings are supposed unital and commutative. The unit group of a ring R isdenoted by R × . Let M be a finitely generated R -module. We denote by µ ( M ) the minimal number of generators of M . For n ≥ µ ( M ) , we denote by V n ( M ) the set of generating vectors of M of length n , i.e., the set of n -tuples in M n whose components generate M . We consider the action of GL n ( R ) on V n ( M ) by matrix right-multiplication. Let SL n ( R ) be the subgroup of GL n ( R ) of thematrices with determinant . Let E n ( R ) be the subgroup of SL n ( R ) generatedby the elementary matrices , i.e., the matrices which differ from the identity bya single off-diagonal element. Let G be a subgroup of GL n ( R ) . Two generatingvectors m , m ′ ∈ V n ( M ) are said to be G -equivalent , which we also denote by m ′ ∼ G m , if there exists g ∈ G such that m ′ = m g . Our chief concern is thedescription of the quotient V n ( M ) /G with G ∈ { SL n ( R ) , E n ( R ) , GL n ( R ) } for R in a class of rings comprising the Dedekind rings. Date : December 11, 2020.2020
Mathematics Subject Classification.
Primary 13E15, Secondary 13F05, 13F05,19B10.
Key words and phrases. elementary divisor ring; semihereditary ring; coherent ring;Prüfer domain; local-global ring; generalized Euclidean ring; special Whitehead group;Nielsen equivalence;
Let us highlight earlier results pertaining to this topic. When M = R ,the elements of V n ( M ) are usually called the unimodular rows of size n andthe orbit set W n ( R ) + V n ( R ) / E n ( R ) has been extensively studied [Lam06,Section VIII.5]. Suslin and Vaserstein [Bas75, Théorème 4] [VS76, Corollary7.4] discovered that W ( R ) can be identified with the elementary symplecticWitt group of R if E n ( R ) acts transitively on V n ( R ) for every n > . In [vdK83,Theorem 3.6], van der Kallen used the former structure inductively to definea structure of Abelian group on W n ( R ) when R has finite stable rank and n is large enough. The interested reader is referred to [DTZ18] for the latestdevelopments on the van der Kallen group structure of W n ( R ) for an affinedomain R . In the latter article, Das, Tikader and Zinna describe moreover theAbelian group structure of V n +1 ( R ) / SL n +1 ( R ) induced by W n +1 ( R ) for R asmooth affine real algebra of dimension n ≥ [DTZ18, Theorem 1.2]; see also[Fas11] for seminal results of this flavor.For R = S [ x , . . . , x m ] , the ring of polynomials in m indeterminates over aring S , Murthy determined conditions under which GL n ( R ) acts transitivelyon V n ( P rk =1 Rx i ) for r ≤ n ≤ m [Mur03].To introduce further known results, let us define the determinant det( M ) of afinitely generated R -module M as the exterior product V µ M where µ = µ ( M ) .The module det( M ) is a cyclic R -module whose annihilator is Fitt µ − ( M ) , the µ -th Fitting ideal of M (see Section 2.1 for definition). The determinant map det : M n → V n M is defined by det( m , . . . , m µ ) = m ∧ · · · ∧ m n . It is imme-diate to check that det(V µ ( M )) ⊆ V (det( M )) . For I an ideal of a ring R , weshowed that V ( I ) / SL ( R ) is equipotent with a subset of V (det( I )) [Guy20,Theorem A] if I can be generated by two elements which are not zero-divisors.For R a quasi-Euclidean ring and M an arbitrary finitely generated R -module,we proved that the determinant map induces a bijection from V µ ( M ) / E µ ( R ) onto V (det( M )) and that E n ( R ) acts transitively on V n ( M ) for every n > µ [Guy17, Theorem A and Corollary B]. In this article we generalize [Guy17,Theorem A and Corollary B] to two classes of rings: the elementary divisorrings and the rings considered by Couchot in [Cou07, Theorems 1 and 2]. Forthe time being, we denote by D the union of these two classes. Their defini-tions are postponed to Section 2.3. The reader should only bear in mind that D contains the Dedekind rings and some natural generalizations.The rings in D share three features which make them particularly amenableto the study of the generating vectors of their finitely presented modules. Forany such ring R , the following hold (see Proposition 2.15 and Section 2.3’stheorems):(1) Every finitely generated ideal of R can be generated by two elements.(2) The stable rank of R is at most . QUIVALENT GENERATING VECTORS OF FINITELY GENERATED MODULES OVER COMMUTATIVE RINGS3 (3) Finitely presented R -modules have an invariant decomposition .Our first result relates SL n ( R ) -orbits of generating vectors to the image ofthe determinant map. Theorem A.
Let R be a ring in D . Let M be a finitely presented R -moduleand set µ = µ ( M ) . Then the determinant map induces a bijection from V µ ( M ) / SL µ ( R ) onto V (det( M )) . In addition, the group SL n ( R ) acts transi-tively on V n ( M ) for every n > µ . In order to generalize Theorem A to direct products with factors in D , weintroduce the following definition. Definition B.
A finitely generated R -module M is said to satisfy the prop-erty ∆ SL if the determinant map induces a bijection from V n ( M ) / SL n ( R ) to V ( V n M ) for every n ≥ µ ( M ) . The ring R is said to satisfy the property ∆ SL if every finitely presented R -module satisfies ∆ SL . We define likewise the property ∆ E for which an analog of Proposition Cbelow holds (see Propositions 5.3 and 5.4). Proposition C (Propositions 5.3 and 5.4) . The two following hold. ( i ) Let { R x } x ∈ X be a family of rings satisfying ∆ SL and let R = Q x R x .Then R satisfies ∆ SL . ( ii ) Let R be a ring which satisfies ∆ SL . Let R be a quotient of R such thatthe natural map SL n ( R ) → SL n ( R ) is surjective for every n . Then R satisfies ∆ SL . Proposition C implies that a direct product of rings, each factor of which isin D or is the quotient of a ring in D , has property ∆ SL .Let us now outline a class of rings R for which the action of SL n ( R ) on V n ( M ) is always transitive. We say that a ring R has property T SL if for everyfinitely presented R -module M , the group SL n ( R ) acts transitively on V n ( M ) for every n ≥ µ ( M ) . We define likewise the property T G for G ∈ { E , GL } . If R satisfies T SL , then the only unit of R is , its identity element. Commutativerings with no units other than have been studied by Cohn [Coh58, Theorem3]. For every Boolean ring B , both B and B [ x ] have no units other than .Since B and B [ x ] are elementary divisor rings [Sho74, Example 2], both enjoyproperty T SL by Theorem A.Regarding the action of GL n ( R ) on V n ( R ) , we observe that it is essentiallydetermined by the action of SL n ( R ) . Indeed, decomposing GL n ( R ) ≃ SL n ( R ) ⋊ R × , we easily see that V n ( M ) / GL n ( R ) = ( R × ) n \ V n ( M ) / SL n ( R ) where ( R × ) n acts on the left via componentwise multiplication. If for instance an R -module M has property ∆ SL , then V n ( M ) / GL n ( R ) identifies with R × \ V (det( M )) ≃ ( R/ Fitt µ − ( M )) × / ( R × + Fitt µ − ( M )) . LUC GUYOT
If a ring R has the property T GL , then obviously every surjective ring homo-morphism R ։ R must induce a surjective homomorphism R × ։ ( R ) × onunit groups. The latter condition is equivalent to say that R has stable rank ,see [EO67, Lemma 6.1] or [Guy20, Proposition 4.2]. Suppose R has property ∆ SL , e.g., R lies in D . Then it is also clear that R has property T GL if R has moreover stable rank . For instance, the ring in [Che17, Example 7] isa Jacobson principal ideal domain of stable rank and thus enjoys property T GL .Besides the action of a subgroup G ⊆ GL n ( R ) on V n ( M ) , it is also naturalto consider the action of Aut R ( M ) , the group of the R -automorphisms of M ,acting diagonally on M n from the left. This action commutes with the actionof G from the right, so that Aut R ( M ) × G acts also on V n ( M ) from the left.The following is an immediate consequence of Theorem A and of the invariantdecompositions of the modules at play (see Theorems 2.9 and 2.12 below): Corollary D.
Let R be a ring in D . Let M be a finitely presented R -module.Then Aut R ( M ) × SL n ( R ) acts transitively on V n ( M ) for every n ≥ µ ( M ) . We finally consider the action of E n ( R ) on generating vectors. Evidently,Theorem A provides us with a description of V n ( M ) / E n ( R ) for n ≥ µ ( M ) ifSL n ( R ) = E n ( R ) . If the latter identity holds, the ring R is termed a GE n -ring .If R is GE n ring for every n ≥ , then R is termed a GE -ring . The class ofGE-rings includes quasi-Euclidean rings, rings of stable rank and Dedekinddomains of arithmetical type with infinitely many units [Vas72]. By takingdirect products of GE-rings, we can obtain non-GE-rings for which E n ( R ) canreplace SL n ( R ) in Theorem A. It is well-known that the class of GE n -ringis stable under the formation of direct products with finitely many factors[Coh66, Theorem 3.1], but not for infinitely many ones. A typical counter-example is Q x Z where x ranges in an infinite set (use, e.g., [Coh66, Lemma5.1]). Although the latter ring is not GE , it is a GE n -ring for every n > .This follows from [CK83, Main Theorem] which implies more generally that anarbitrary direct product of rings of integers or their localizations, is a GE n -ringfor every n > . For such products, combining Theorem A with PropositionC yields a description of V µ ( M ) / E µ ( R ) provided that µ = µ ( M ) > .If R is not a GE n -ring, describing V n ( M ) / E n ( R ) is already challengingwhen we restrict to M = R and n = 2 . In this case, V ( R ) / E ( R ) identi-fies with E ( R ) \ SL ( R ) / E ( R ) where E ( R ) is the group of matrices of theform (cid:18) r (cid:19) with r ∈ R (see Lemma 4.4). If for instance R is the ring ofintegers of an imaginary quadratic field Q ( √− d ) that is not GE , then it fol-lows from [FF88, Theorem 2.4] together with the Normal Form Theorem forFree Products with Amalgamation [LS77, Theorem IV.2.6] that V ( R ) / E ( R ) QUIVALENT GENERATING VECTORS OF FINITELY GENERATED MODULES OVER COMMUTATIVE RINGS5 is infinite; see [Nic11, Theorem 1.5] and [She16] for an elementary proof of thisfact. The quotient V ( R ) / E ( R ) clearly surjects on the quotient of SL ( R ) bythe subgroup of unipotent matrices. For infinitely many d > , this quotientsurjects in turn on a non-Abelian free group [GMV94, Proof of Theorem 1.2].With our last result we depart from such intricate situations, assuming that µ ( M ) > and that M has a non-trivial torsion submodule. Under these as-sumptions, we can describe V n ( M ) / E n ( R ) by means of the determinant mapand the special Whitehead group SK ( R ) of R (see Section 5 for the definitionof SK ). Let us denote by τ ( M ) the torsion submodule of M , that is, τ ( M ) + { m ∈ M | rm = 0 for some r ∈ R which is not a zero divisor } . A ring R is a Bézout ring if its finitely generated ideals are principal.
Theorem E.
Assume that R is a Bézout ring whose proper quotients havestable rank or that R is an almost local-global coherent Prüfer ring (seeSection 2.3 for definitions). Let M be a finitely presented R -module. Let e be an idempotent of R such that µ ( eM ) = µ ( M ) and set µ ′ = µ ( τ ( M )) , µ = µ ( M ) . Then the following hold. ( i ) If µ ′ > or µ = 1 , then the determinant map induces a bijection fromfrom V µ ( M ) / E µ ( R ) onto V (det( M )) . ( ii ) If µ ′ = 1 and µ > , then V µ ( M ) / E µ ( R ) is equipotent with V (det( M )) × SK ( R ) . ( iii ) If either µ ′ = 0 , or µ ′ = 0 and eM surjects onto a non-principalinvertible ideal of Re , then E n ( R ) acts transitively on V n ( M ) for every n > µ . This holds in particular if R is an almost local-global Prüferdomain and M is torsion-free but not free. ( iv ) The group E n ( R ) acts transitively on V n ( M ) for every n > µ + 1 . A Bézout ring R such that the proper quotients of R/ J ( R ) have stablerank is an elementary divisor ring [McG08, Proposition 3.5 and Theorem3.7]. To the best of our knowledge, all known examples of elementary divisorrings satisfy this property (this is corroborated by [McG08, Remark 4.7]).Combining Theorem E with Proposition 2.7 below, we gain thus insight on allsuch examples. Final Remarks.
The proof Theorem A hinges on the fact that ideals aretwo-generated for rings in D (see specifically the results of Section 3). Theclass of Bass rings is another well-studied class of rings whose ideals are two-generated [Lam00, §4] and whose finitely generated torsion-free modules arewell-understood [LW85]. The Bass rings constitute therefore a class of choiceto pursue the study of equivalent generating vectors in the spirit of TheoremsA and E. LUC GUYOT
Layout.
This paper is organized as follows. Section 2 introduces notationand background results. It encloses some of the fundamental properties of thedeterminant map, of the Bass stable rank and of the class D . Section 3 isdedicated to the study of the action of SL n ( R ) on V n ( I ) for I a two-generatedideal of R . Section 4 addresses the finitely generated modules which admit aninvariant decomposition in the sense of Theorems 2.9 and 2.12. However, theidempotents appearing in the decomposition of Theorem 2.12 are only handledin Section 5 where Proposition C is proven. Section 6 is dedicated to the proofsof the Theorems A and E. Acknowledgments.
We are grateful to François Couchot and Jean Fasel fortheir encouragements. We are thankful to Justin Chen, Henri Lombardi, Bog-dan Nica and Wilberd van der Kallen for helpful comments and references.2.
Notation and background
We assume throughout that R is a commutative unital ring and M is afinitely generated R -module. We denote by J ( R ) the Jacobson radical of R ,that is, the intersection of the maximal ideals of R . If p is a prime ideal of R , we denote by R p the localization of R at p . Similarly, we denote by M p itslocalization at p , which we identify with M ⊗ R R p . Given an element m in M ,we abuse notation in denoting also by m its image m ⊗ R ∈ M p . If M and N are two submodules of a given R -module, we denote by ( M : N ) the idealconsisting of all elements r ∈ R such that rN ⊆ M . We define the annihilator ann R ( M ) of M as (0 : M ) . If Rm and Rm ′ are two cyclic submodules of M ,we simply write ( m : m ′ ) for ( Rm : Rm ′ ) . Regular elements.
An element r of a ring R is a zero divisor if ann R ( r ) + (0 : r ) = { } . An element of R is regular if it is not a zero divisor. An idealof R is regular if it contains a regular element. The total ring of quotients K ( R ) of R is the localization R [ S − ] where S is the set of regular elementsof R . The above definition of a regular element should not be confused withthe definition of a von Neumann regular element of R , that is an element forwhich there exists a ∈ R satisfying r = ara . A ring whose elements are vonNeumann regular is called a von Neumann regular ring . Matrices over R and M . Given two × n row vectors r = ( r , . . . , r n ) ∈ R n and m = ( m , . . . , m n ) ∈ M n , we denote by r ⊤ and m ⊤ the n × columnvectors obtained by transposition and we define their product rm ⊤ = mr ⊤ + P i r i m i . Based on these identities, the product of any two matrices over R and M , with compatible numbers of rows and columns, is uniquely defined.We denote respectively by m × n and by n the m × n zero matrix and the n × n identity matrix over R . QUIVALENT GENERATING VECTORS OF FINITELY GENERATED MODULES OVER COMMUTATIVE RINGS7
Fitting ideals and the determinant module.
The invariant on whichour study of the action of SL µ ( R ) on V µ ( M ) hinges is the determinant map.This map can be effectively computed thanks to the µ th-Fitting ideal. Fitting ideals.
Let F ϕ −→ G → M → be an exact sequence where F and G are free R -modules and G is finitelygenerated. Let I i ( ϕ ) the ideal of R generated by the i × i minors of the matrixof ϕ : F → G , agreeing that I ( ϕ ) = R . Then the ( i + 1) -th Fitting idealof M is Fitt i ( M ) + I µ ( G ) − i ( ϕ ) . The Fitting ideals are independent of ϕ byFitting’s lemma [Eis95, Corollary 20.4]. They commute with base change , i.e, Fitt i ( M ⊗ R S ) = Fitt i ( M ) ⊗ R S for any R -algebra S [Eis95, Corollary 20.5],and hence with localization. The determinant of a finitely generated module.
Let µ = µ ( M ) . Re-call that the determinant det( M ) of M is the exterior product V µ M . Thisis a cyclic R -module whose annihilator is Fitt µ − ( M ) [Eis95, Exercise 20.9.i].Given m = ( m , . . . , m µ ) ∈ V µ ( M ) , we denote by φ m : det( M ) → R/ Fitt µ − ( M ) the isomorphism induced by the map m ∧ · · · ∧ m µ → µ − ( M ) . Givenanother generating pair m ′ = ( m ′ , . . . , m ′ µ ) ∈ M µ , we definedet m ( m ′ ) + φ m ( m ′ ∧ · · · ∧ m ′ µ ) . It is easy to check that det m (V µ ( M )) is a subgroup of ( R/ Fitt µ − ( M )) × whichdoesn’t depend on m . The following lemma is straightforward. Lemma 2.1.
Let m , m ′ , m ′′ ∈ V µ ( M ) . Then the following assertions hold. ( i ) det m ( m ′ A ) = det m ( m ′ ) det( A ) , for every µ × µ matrix A over R . ( ii ) det m ( m ′′ ) = det m ( m ′ ) det m ′ ( m ′′ ) . ( iii ) det m ( m ′ ) = 1+Fitt µ − ( M ) , if and only if, det m ( m ′ ) = 1+Fitt µ − ( M p ) for every maximal ideal p of R , where m and m ′ denote (abusively)their natural images in V µ − ( M p ) in the left-hand side of the identity. (cid:3) When there is no risk of ambiguity, we simply denote by the identityelement of R/ Fitt µ − ( M ) . The ideal Fitt µ − ( M ) has a convenient descriptionwhich makes the computation of det m effective when a workable presentationof M is given. Let m ∈ V µ ( M ) . We say that an element r ∈ R is involved in arelation of m if there is ( r , . . . , r µ ) ∈ R µ such that P µi =1 r i m i = 0 and r = r i for some i . Lemma 2.2.
Let m ∈ V µ ( M ) . Then Fitt µ − ( M ) is the set of the elements of R involved in a relation of m . (cid:3) LUC GUYOT
Let m = π ( m ) be the image of m ∈ V µ ( M ) by the natural map π : M → M/ Fitt µ − ( M ) M and let e be the canonical basis of ( R/ Fitt µ − ( M )) µ . Thenthe map m e induces an isomorphism ϕ m from M/ Fitt µ − ( M ) M onto ( R/ Fitt µ − ( M )) µ . The next lemma shows how the map π can be used tocompute det m . Lemma 2.3.
Let m , m ′ ∈ V µ ( M ) . Then det m ( m ′ ) is the determinant of theunique µ × µ matrix A over R/ Fitt µ − ( M ) satisfying π ( m ′ ) = π ( m ) A , that isthe matrix of ϕ m ◦ π ( m ′ ) with respect to e . (cid:3) Ranks.
Following [Rie83, Section 9] and [MR87, Section 6.7.2], we definethe Bass stable rank of a finitely generated R -module M . An integer n > lies in the stable range of M if for every m = ( m , . . . , m n +1 ) ∈ V n +1 ( M ) ,there is ( r , . . . , r n ) ∈ R n such that ( m + r m n +1 , . . . , m n + r n m n +1 ) belongsto V n ( M ) . If n lies in the stable range of M , then so does k for every k > n [MR87, Lemma 11.3.3]. The stable rank sr ( M ) of M is the least integer in thestable range of M .The Bass stable rank can be characterized in terms of a lifting property forthe generating vectors of quotient modules. Proposition 2.4.
Let M be a finitely generated R -module and let n ≥ µ ( M ) .Then the following are equivalent: ( i ) sr ( M ) ≤ n . ( ii ) For every R -submodule N ⊆ M , the map V n ( M ) → V n ( M/N ) sending ( m , . . . , m n ) to ( m + N, . . . , m n + N ) is surjective. In other words,every generating n -vector of M/N lifts to a generating n -vector of M .Proof. Straightforward, see the proof of [Guy20, Proposition 4.2]. (cid:3)
Extending naturally the ranks introduced in [MR87, Section 11.3.3] to finitelygenerated modules, we define the linear rank glr R ( M ) and the elementary rank er R ( M ) of M as the least integer n ≥ µ ( M ) such that GL k ( R ) , respectivelyE k ( R ) , acts transitively on V k ( M ) for every k > n . Note that the analogousdefinition based on SL k ( R ) yields the rank glr R ( M ) . When there is no risk ofambiguity, we simply write glr ( M ) and er ( M ) instead of glr R ( M ) and er R ( M ) .It is easy to check that SL ( R ) acts transitively on V ( R ) . Hence glr ( R ) = 2 is equivalent to glr ( R ) = 1 . A ring R is said to be a Hermite ring if glr ( R ) = 1 .This is [Lam06, Definition I.4.6]; Lam’s Hermite rings are sometimes called completable rings.Following [Coh66], we call R a GE n -ring if SL n ( R ) = E n ( R ) and R is saidto be a generalized Euclidean ring , or simply a GE -ring , if it is a GE n -ring forall n > . QUIVALENT GENERATING VECTORS OF FINITELY GENERATED MODULES OVER COMMUTATIVE RINGS9
Remark 2.5.
The ring R is a GE -ring if and only if E ( R ) acts transitivelyon V ( R ) . If R is Hermite, then R is a GE -ring if and only if er ( R ) = 1 . We denote by J ( M ) , the Jacobson radical of M , that is, the intersection ofthe maximal submodules of M (see [Lam01, §24] for properties and examples). Proposition 2.6. [MR87, Proposition 11.3.11 and Lemma 11.4.6]
Let M be afinitely generated R -module and let N be a submodule of M . Then the followinghold. ( i ) er ( R ) ≤ sr ( R ) . ( ii ) µ ( M ) ≤ glr ( M ) ≤ er ( M ) and sr ( M ) ≤ µ ( M ) + sr ( R ) − . ( iii ) sr ( M/N ) ≤ sr ( M ) and equality holds if N ⊆ J ( M ) . Proposition 2.7.
Let M be a finitely generated R -module. Then the followinghold. ( i ) er ( M ) = er R/I ( M ) for every ideal I ⊆ ann( M ) . ( ii ) glr ( M ) = glr R/I ( M ) for every ideal I ⊆ ann( M ) ∩ J ( R ) . ( iii ) If M = { } , then er ( M ) ≤ µ ( M ) + sr ( M ) − . ( iv ) glr ( M/N ) = glr ( M ) and er ( M/N ) = er ( M ) for every R -submodule N ⊆ J ( M ) . ( v ) If M = { } and sr ( R ) = 1 then er ( M ) = sr ( M ) = µ ( M ) .Proof. ( i ) . Observe that the natural map E n ( R ) → E n ( R/I ) is surjective forevery ideal I of R and every n > . The result follows immediately. ( ii ) .Observe that the natural map GL n ( R ) → GL n ( R/I ) is surjective for everyideal I ⊆ J ( R ) and every n > [Wei13, Exercise I.1.12.iv]. ( iii ) . We can assume, without loss of generality, that sr ( M ) < ∞ . Let n >µ ( M ) + sr ( M ) − and let m ∈ V n ( M ) , m ′ = ( m ′ , . . . , m ′ µ , , . . . , ∈ V n ( M ) with µ + µ ( M ) . Let s + sr ( M ) . It readily follows from the definition of s that m ∼ E n ( R ) ( m ′′ , . . . , m ′′ s , , . . . , for some ( m ′′ , . . . , m ′′ s ) ∈ V s ( M ) . Thefollowing equivalences are then immediate. m ∼ E n ( R ) ( m ′′ , . . . , m ′′ s , m ′ , . . . , m ′ µ , , . . . , , ∼ E n ( R ) (0 , . . . , , m ′ , . . . , m ′ µ , , . . . , , ∼ E n ( R ) m ′ . This shows that er ( M ) < n and hence er ( M ) ≤ µ + s − . ( iv ) . The inequality glr ( M ) ≥ glr ( M/N ) is trivial. To prove the reverseinequality, let µ + µ ( M ) = µ ( M/N ) and n > glr ( M/N ) . Let m ∈ V n ( M ) and m ′ = ( m ′ , . . . , m ′ µ , , . . . , ∈ V n ( M ) . By hypothesis, we can find n =( ν , . . . , ν n ) ∈ N n such that m ∼ GL n ( R ) m ′ + n . As ( m ′ + ν , . . . , m ′ µ + ν µ ) generates M , we easily derive that m ′ + n ∼ GL n ( R ) m ′ . Thus we established that m ∼ GL n ( R ) m ′ for every m ∈ V n ( R ) , which proves that glr ( M ) ≤ glr ( M/N ) .The proof of the analog result for the elementary rank is identical. ( v ) . The equality sr ( M ) = µ ( M ) is given by Proposition 2.6. ii . We shallprove that er ( M ) = µ ( M ) by induction on µ + µ ( M ) > . If µ = 1 , then M ≃ R/ ann( M ) . As er ( M ) = er R/ ann( M ) ( M ) by ( i ) , we deduce from Proposition2.6 that er ( M ) ≤ sr ( R/ ann( M )) ≤ sr ( R ) . Therefore er ( M ) = 1 . Assume nowthat µ > and let n > µ, m ∈ V n ( M ) and m ′ = ( m ′ , . . . , m ′ µ , , . . . , ∈ V n ( M ) . Since n > µ = sr ( M ) , we have m ∼ E n ( R ) ( m , . . . , m µ , m ′ , , . . . , for some ( m , . . . , m µ ) ∈ V µ ( M ) . The induction hypothesis applied to M/Rm ′ yields m ∼ E n ( R ) ( m ′ + λ m ′ , . . . , m ′ µ + λ µ m ′ , m ′ , , . . . , for some λ , . . . , λ µ ∈ R . Therefore m ∼ E n ( R ) m ′ . (cid:3) A ring R is semi-local if it has only finitely many maximal ideals. A ring R is K-Hermite if for every n ≥ and every r ∈ R n , there is γ ∈ GL n ( R ) such that r γ = (0 , . . . , , d ) for some d ∈ R . It follows immediately from thedefinition that a K-Hermite ring is both a Bézout ring and a Hermite ring. Proposition 2.8.
The following hold: ( i ) sr ( R ) = 1 if R is semi-local [Bas64, Corollary 10.5] . ( ii ) sr ( R ) ≤ if R is a K-Hermite ring [MM82, Proposition 8. i ] . ( iii ) sr ( R ) ≤ if every proper quotient of R/ J ( R ) has stable rank [McG08,Theorem 3.6] . ( iv ) sr ( R ) ≤ dim Krull ( R ) + 1 [Hei84, Corollary 2.3] or [CLQ04, Theorem2.4] . Elementary divisor rings and coherent Prüfer rings.
In this sec-tion, we underscore the properties of the rings in D which are instrumentalin the proofs of Theorems A and E. We detail in particular the structure offinitely presented modules over these rings by stating the corresponding in-variant decomposition theorems. Complete definitions are provided and someremarkable examples and results are briefly outlined. Recall that the class D is the union of two classes: the elementary divisor rings and the almostlocal-global coherent Prüfer rings ; the latter are the almost local-global semi-hereditary rings of Couchot [Cou07].A ring R is an elementary divisor ring if every matrix over R with finitelymany rows and columns admits a Smith Normal Form, that is, for every m × n matrix A over R , we can find B ∈ GL m ( R ) , C ∈ GL n ( R ) such that BAC isa diagonal matrix whose diagonal coefficients d , . . . , d k satisfy the divisibilitycondition d i +1 | d i for every ≤ i ≤ k − .The class of elementary divisor rings is easily seen to be stable under theformation of direct products and quotients. An elementary divisor ring isclearly a K-Hermite ring (see definition right before Proposition 2.8), hence QUIVALENT GENERATING VECTORS OF FINITELY GENERATED MODULES OVER COMMUTATIVE RINGS11 a Hermite ring. A finitely presented module over an elementary divisor ringdecomposes into a direct sum of finitely many cyclic modules whose annihilatorforms a descending chain.
Theorem 2.9. [Kap49, Theorem 9.1]
Let R be an elementary divisor ring.Then R satisfies the Invariant Factor Theorem, that is, every finitely presented R -module M decomposes as R/ a × · · · × R/ a k ( k ≥ where the ideals a i ⊆ R form a descending chain R ) a ⊇ · · · ⊇ a k . Conversely, a ring whose finitely presented modules have a decompositioninto a direct sum of cyclic modules is an elementary divisor rings [LLS74,Theorem 3.8].
Examples 2.10.
Principal ideal rings, i.e., rings whose ideals are principal,are elementary divisor rings [Kap49, Theorem 12.3] . The ring of entire func-tions and the ring of algebraic integers are two emblematic non-Noetherian el-ementary divisor domains [DB72, Examples 1 and 2 of Section 4] . A Booleanring and more generally a von Neumann ring R [GH56, Remark 12] as well as R [ X ] [Sho74, Example 2] are elementary divisor rings. We turn now to the definition of an almost local-global coherent Prüfer ring .A ring is reduced if it has no non-zero nilpotent element. A ring is arith-metic if its finitely generated ideals are locally principal. A
Prüfer ring inthe sense of Hermida and Sànchez-Giralda [HSG86, Definition 4] is a reducedarithmetical ring [LQ15, Proposition and definition VIII.4.4]. A ring is coher-ent if its finitely generated ideals are finitely presented. A Prüfer coherent ringis characterized by the property that its finitely generated ideals are projective[LQ15, Theorem XII.4.1]. A Prüfer coherent ring is often called a semiheredi-tary ring, a terminology we shall discontinue (following Lombardi and Quitté[LQ15]) as we find it not suggestive enough.A ring R is local-global , or LG for short, if every (possibly multivariate)polynomial over R whose values generate R , represents a unit. The LG rings[MW81, EG82] are a natural generalization of the rings satisfying van derKallen’s primitive criterion [vdK77, Definition 1.10], see [MW81, Propositionon page 456]. Let f ( X ) = aX + b with ( a, b ) ∈ V ( R ) . Clearly, the idealgenerated by the values f ( R ) of f is R . If R is LG, then there is x ∈ R suchthat ax + b is a unit. Therefore LG rings have stable rank . A ring R is almostlocal-global in the sense of Couchot [Cou07], or almost-LG for short, if R/Rr is LG for every regular element r ∈ R . Examples 2.11.
Semi-local rings and rings of Krull dimension (e.g. vonNeumann regular rings) are LG rings [FS01, Examples 4.1 and 4.2 of Chapter V] , [LQ15, Fact IX.6.2] . The Nagata ring R ( X ) of a ring R is an LG ring [LQ15, Fact IX.6.7] . A Dedekind domain is a Noetherian Prüfer domain. Dedekind domainsare one-dimensional [Mat89, Theorem 11.6], hence almost-LG. The classicalstructure theorem of Steinitz for finitely generated modules over Dedekindrings [Mag02, Theorem 7.48] generalizes to almost-LG coherent Prüfer ringsin the following way.
Theorem 2.12. [Cou07, Theorem 2]
Let R be an almost-LG coherent Prüferring. Then R satisfies the Invariant Factor Theorem, that is, every finitelypresented R -module M decomposes as M ≃ m Y i =1 R/I i × n Y j =1 (cid:0) J j, e j × · · · × J j,k j e j (cid:1) . where R ) I ⊇ · · · ⊇ I m = { } are invertible ideals of R , ( e , . . . , e n ) is asequence of orthogonal idempotents, ( k , ..., k n ) a strictly increasing sequenceof positive integers and each ideal J j,k e j is an invertible ideal of Re j . In ad-dition, the isomorphism class of M is completely determined by the followinginvariants: • the ideals I , . . . , I m , • the idempotents e , . . . , e n , • the integers k , . . . , k n and • the isomorphism class of ( J , · · · J ,k e ) × · · · × ( J n, · · · J n,k n e n ) . Remark 2.13.
Theorem 2.12 implies in particular that J j, e j × · · · × J j,k j e j ≃ ( J , · · · J ,k j e j ) × ( Re j ) k j − for every ≤ j ≤ n . A ring R is said to be of finite character if R/I is semi-local for everynon-zero ideal I of R . Using Examples 2.11, it is immediate to check that aPrüfer domain which is one-dimensional or of finite character is an almost-LGcoherent Prüfer ring but not necessarily Noetherian. Examples 2.14.
The ring of algebraic integers is a non-Noetherian Bézoutdomain (see Examples 2.10) which is one-dimensional by the going-up andgoing-down theorems [Mat89, Theorem 9.4] . Let p be a prime number, let Z p be the ring of the p -adic integers and let Q p be its fraction field. Thenthe integral closure of Z p in an algebraic closure of Q p is a one-dimensionalvaluation domain, with value group Q , hence not Noetherian [Mat89, Theorem11.1] . We conclude this section with two properties of importance for the proofsof our theorems.
QUIVALENT GENERATING VECTORS OF FINITELY GENERATED MODULES OVER COMMUTATIVE RINGS13
Proposition 2.15.
Let R be a ring in D . Then the two following hold. ( i ) Every finitely generated ideal of R can be generated by two elements. ( ii ) The stable rank of R is at most .Proof. Assertion ( i ) is obvious for a Bézout ring, hence for any elementarydivisor ring. If R is an almost-LG coherent Prüfer ring, then ( i ) follows from[Cou07, Theorem 2 and Lemma 10]. Assertion ( ii ) for an elementary divisorring is given by Proposition 2.8. ii . For an almost-LG coherent Prüfer ring,combine Proposition 3.8 below and [Cou07, Lemma 10]. (cid:3) The quotient V n ( I ) / SL n ( R ) for n ≥ In this section, I denotes a two-generated ideal of R . Our results assumethat I enjoys at least one of the following two properties. Definition 3.1.
Let I be an ideal of R and let I − + { x ∈ K ( R ) | xI ⊆ R } where K ( R ) is the total ring of quotients of R . • The ideal I is said to be invertible if II − = R . • The ideal I is said to be / -generated if I/Rx is generated by oneelement for every x ∈ R \ { } . It is well-known that an invertible ideal contains a regular element and thatit is projective of constant rank (see, e.g., [Eis95, Theorem 11.6], wherethis claim is proved in 11.6.c and 11.6.d without actually assuming that R is Noetherian). A / -generated ideal which contains a regular element isinvertible [LM88, Theorem 1]. In a one-dimensional domain [Hei76, Theorem3.1] or in a ring of finite character [GH70, Theorem 3], invertible ideals are / -generated. Lemma 3.2.
Let I and a be ideals of R with I invertible. If I/I a is cyclicthen I/I a ≃ R/ a .Proof. Since
I/I a is cyclic, it suffices to show that ann( I/I a ) = a . Clearly ann( I/I a ) = ( I a : I ) and the inclusions ( I a : I ) ⊇ a and ( I a : I ) ⊆ ( II − a : II − ) are immediate. To conclude, we observe that ( II − a : II − ) = a holdsbecause I is invertible. (cid:3) Lemma 3.3.
Let I be a / -generated ideal of R . Assume that at least one ofthe following holds. ( i ) Every proper quotient of R has stable rank . ( ii ) Every proper quotient of R/ J ( R ) has stable rank and I is invertible.Then sr ( I ) ≤ . In particular sr ( R ) ≤ and R is Hermite. Proof.
Let ( a, b, c ) ∈ V ( I ) . We need to prove the existence of ( r , r ) ∈ R such that ( a + r c, b + r c ) ∈ V ( I ) .We assume first that ( i ) holds. The proof follows closely the lines of [McG08,Theorem 3.6]. If a = 0 , then we can take ( r , r ) = (1 , . Thus we cansuppose that a = 0 . If I = Ra , then we can take ( r , r ) = (0 , . Thuswe can suppose that Ra = I . As I is / -generated, the non-empty set X of the ideals I ′ verifying Ra ⊆ I ′ ( I is isomorphic to the set of properideals of R containing ( a : I ) as a set partially ordered by inclusion. Hencewe can consider the intersection J of the maximal elements of X . Since J contains a , it is a non-zero ideal. Thus sr ( I/J ) = 1 , so that we can find r ∈ R satisfying b + rc + J = I . Reasoning by contradiction, we assumethat Ra + R ( b + rc ) ( I . Then Ra + R ( b + rc ) is contained in a maximalelement K ∈ X . As a result, we have J ⊆ K ( I . We obtain a contradictionby observing that K ⊇ Ra + R ( b + rc ) + J = I . Therefore we can take ( r , r ) = (0 , r ) .Let us assume now that ( ii ) holds. If I J ( R ) = { } , then J ( R ) = { } so that ( i ) is satisfied. Otherwise, we have I/ J ( R ) I ≃ R/ J ( R ) by Lemma3.2. Hence sr ( I/ J ( R ) I ) ≤ by ( i ) . The result follows then from Proposition2.6. iii . (cid:3) Proposition 3.4.
Let R and I be as in Lemma 3.3. Then er ( I ) ≤ .Proof. Let ( a , . . . , a n ) ∈ V n ( I ) with n > . If any of ( i ) or ( ii ) holds, thensr ( I ) = 2 by Lemma 3.3. Hence we can assume, without loss of generality,that n = 3 and a = 0 .We assume first that ( i ) holds. Let b ∈ I be such that I = Ra + Rb . Thenthe following equivalence holds(1) ( a , a , a ) ∼ E ( R ) ( a , b, . Indeed, since sr ( I/Ra ) = sr ( R/ ( a : I )) = 1 , we have ( a + Ra , a + Ra ) ∼ E ( R ) ( b + Ra , Ra ) from which we easily infer the equivalence (1). Fixnow ( a ′ , a ′ , ∈ V ( I ) with a ′ = 0 and let b ′ ∈ I be such that I = Ra ′ + Rb ′ .Then we have(2) ( a , b, ∼ E ( R ) ( a ′ , a , b ) while the two equivalences(3) ( a ′ , a , b ) ∼ E ( R ) ( a ′ , b ′ , , ( a ′ , a ′ , ∼ E ( R ) ( a ′ , b ′ , can be derived from (1) using obvious substitutions. Combining (1), (2) and(3) yields ( a , a , a ) ∼ E ( R ) ( a ′ , a ′ , , which proves the result.Let us now assume that ( ii ) holds. We can also assume, without loss of gen-erality, that I J ( R ) = { } , since otherwise ( i ) is satisfied. Thus I/I J ( R ) ≃ QUIVALENT GENERATING VECTORS OF FINITELY GENERATED MODULES OVER COMMUTATIVE RINGS15 R/ J ( R ) by Lemma 3.2. As we have er ( I ) = er ( I/I J ( R )) = er R/ J ( R ) ( I/I J ( R )) by Proposition 2.7, the result follows from ( i ) . (cid:3) Lemma 3.5.
Let I be an invertible ideal of R . Suppose there is a ∈ I suchthat a + J ( R ) is contained in only finitely maximal ideals of R . Then there is b ∈ I such that I = Ra + Rb . Lemma 3.5 implies that an invertible ideal I of R is generated by / elementif R/ J ( R ) is of finite character. This lemma is a straightforward generalizationof [GH70, Theorem 3]. We provide a proof for the convenience of the reader. Proof. If a = 0 , then R is semi-local so the result holds by [GH70, Corollary2]. If I = Ra , the result is obvious. Otherwise b + ( a : I ) is a proper idealof R and satisfies Ra = I b . Since Ra + J ( R ) ⊆ b + J ( R ) , there are onlyfinitely many maximal ideals p , . . . , p n of R containing b + J ( R ) . By [GH70,Theorem 2], we can find b ∈ I \ S ni =1 I p i . We have Rb = I c for c = ( b : I ) and c + J ( R ) is not contained in any of the ideals p i . Therefore Ra + Rb = I ( b + c ) and R = b + c + J ( R ) as the latter ideal is not contained in any maximal idealof R . Thus I = Ra + Rb + J ( R ) I and the result follows from Nakayama’sLemma [Eis95, Corollary 4.8]. (cid:3) Corollary 3.6.
Assume that every proper quotient of R/ J ( R ) has stable rank . Let I be an invertible / -generated ideal of R . Then SL n ( R ) acts transi-tively on V n ( I ) for every n ≥ .Proof. The group E n ( R ) acts transitively on V n ( I ) for every n ≥ by Proposi-tion 3.4. Since I is invertible, it is a faithful projective module of constant rank . By [Guy20, Corollary 3.2], the group SL ( R ) acts transitively on V ( I ) . (cid:3) Corollary 3.7.
Assume that R/ J ( R ) is of finite character. Let I be an in-vertible ideal of R . Then SL n ( R ) acts transitively on V n ( I ) for every n ≥ .Proof. Combine Lemma 3.3, Lemma 3.5 and Corollary 3.6. (cid:3)
A ring R is a P P -ring if every principal ideal of R is projective. A co-herent Prüfer ring is, for instance, a PP-ring. Coherent Prüfer rings can becharacterized as the arithmetic PP-rings [LQ15, Theorem XII.4.1]. Proposition 3.8.
Assume R is a P P -ring and let I be an invertible ideal of R . Suppose moreover that the two following hold: (i) sr ( R/Rr ) = 1 for every regular element r ∈ R . (ii) µ ( I/Rr ) = 1 for every regular element r ∈ I .Then I is generated by two regular elements and we have: • sr ( I ) ≤ , • er ( I ) ≤ . In particular, we have sr ( R ) ≤ and R is Hermite. In addition, the group SL n ( R ) acts transitively on V n ( I ) for every n ≥ .Proof. Since I is invertible, it contains a regular element x . Using hypothesis ( ii ) , we infer that I = Rx + Rx ′ for some x ′ ∈ R . As R is a P P -ring, it isadditively regular [Mat83, Propositions 1.4 and 2.8]. Hence we can find λ ∈ R such that y + x + λx ′ is regular. Thus I is generated by the two regularelements x and y .Let us prove first that sr ( I ) ≤ . To do so, consider ( a, b, c ) ∈ V ( I ) . If a = 0 , we can replace, without loss of generality, a with x . In this case, we cansuppose a regular so that sr ( I/Ra ) = sr ( R/ ( a : I )) = 1 . Consequently, thereis r ∈ R such that b + rc + Ra generates I/Ra , i.e., ( a, b + rc ) ∈ V ( I ) . If a is not regular, then (0 : a ) = Re for some idempotent e ∈ R . Applying theprevious reasoning to Ie and I (1 − e ) in the PP-rings Re and R (1 − e ) yields ( r , r ) ∈ R such that ( a + r c, b + r c ) ∈ V ( I ) .Let us prove now that er ( I ) ≤ . Let ( a, b, c, a . . . , a n − ) ∈ V n ( I ) with n > . Because sr ( I ) ≤ , we can assume without loss of generality that n = 3 and c = x . Using the fact that sr ( I/Rx ) = sr ( R/ ( x : I )) = 1 , we deducethat ( a + Rx, b + Rx ) ∼ E ( R ) (0 , y + Rx ) and hence ( a, b, c ) ∼ E ( R ) ( x, y, .Therefore er ( I ) ≤ .As SL ( R ) acts transitively on V ( I ) by [Guy20, Corollary 3.2], we deducethat SL n ( R ) acts transitively on V n ( I ) for every n ≥ . (cid:3) Direct products of cyclic modules and ideals
By Theorems 2.9 and 2.12, an R -module M as in Theorems A and E de-composes into a direct product of cyclic modules and ideals. In this section,we describe the action of G ∈ { SL n ( R ) , E n ( R ) } on the generating vectors ofsuch a direct product under assumptions which are less restrictive than thoseof Theorems A and E.Let a , . . . , a k be ideals of R . We denote by e i ( ≤ i ≤ k ) the element ofthe direct product R/ a × · · · × R/ a k whose i -th component is the identity of R/ a i and whose other components are zero. If m = ( m , . . . , m n ) ∈ ( M × N ) n with M = R/ a × · · · × R/ a k and N a finitely generated module, we let Mat( m ) + (( m j ) i ) be the ( k + 1) × n matrix whose columns are the elements m j = (( m j ) i ) with ( m j ) k +1 ∈ N and ( m j ) i ∈ R/ a i for ≤ i ≤ k and ≤ j ≤ n . Clearly, we have Mat( m A ) = Mat( m ) A for every n × n matrix A over R .4.1. Products with two factors.
This section is dedicated to modules of theform R/ a × I ( a , I ⊆ R ). These modules are used as bases in each inductionproof of Section 4.2. QUIVALENT GENERATING VECTORS OF FINITELY GENERATED MODULES OVER COMMUTATIVE RINGS17
Lemma 4.1.
Let M be a finitely generated R -module and let b be an ideal of R contained in ann( M ) . Let m = (( m , , . . . , ( m n − , , ( m n , ∈ V n ( M × R/ b ) with n ≥ µ ( M ) . Then m ∼ E n ( R ) (( m , , . . . , ( m n − , , (0 , .Proof. It suffices to show that m n lies in the submodule of M generated by m , . . . , m n − . Since m generates M × R/ b , we can find r = ( r , . . . , r n ) ∈ R n such that mr ⊤ = ( m n , . We infer from the previous identity that r n ∈ b andhence m n = P n − i =1 r i m i . (cid:3) Lemma 4.2.
Let M = R/ a × R with a an ideal of R . Let m ∈ V ( M ) .Then a = Fitt ( M ) and m ∼ SL ( R ) (det e ( m ) e , e ) where e = ( e , e ) with e = (1 , and e = (0 , . In particular, the map det e induces a bijectionfrom V ( M ) / SL ( R ) onto ( R/ a ) × .Proof. The fact that a = Fitt ( M ) is a direct consequence of Lemma 2.2. Bydefinition, there is a × matrix A over R such that m = e A . Since SL ( R ) acts transitively on V ( R ) , we can find σ ∈ SL ( R ) such that Aσ = (cid:18) u v (cid:19) for some u, v ∈ R . As m σ generates M , Lemma 4.1 applies with b = { } sothat m ∼ SL ( R ) ( ue , e ) . Therefore det e ( m ) = u + Fitt ( M ) = u + a , whichcompletes the proof. (cid:3) Proposition 4.3.
Let M be a two-generated faithful R -module such that M p is isomorphic to R p / a p × R p , for every maximal ideal p of R , with a p someideal of R p . Let m , m ′ ∈ V ( M ) . Then the following are equivalent. ( i ) m ∼ SL ( R ) m ′ , ( ii ) det m ( m ′ ) = 1 .Proof. The implication ( i ) ⇒ ( ii ) follows from Lemma 2.1. i . Let us prove ( ii ) ⇒ ( i ) . By [Guy20, Proposition 3.1] and Lemma 2.1. iii , we can assumethat R is local. Applying Lemma 4.2 and Lemma 2.1. ii yields the result. (cid:3) Given an ideal a of R , we denote by E ( a ) the group of matrices of the form (cid:18) r (cid:19) with r ∈ a . Lemma 4.4.
Let M = R/ a × R with a an ideal of R and let e + ( e , e ) . Forevery m ∈ V ( M ) , fix σ m ∈ SL ( R ) such that m = (det e ( m ) e , e ) σ m . Thenthe two following hold. ( i ) The map ∆ e : m E ( R ) (det e ( m ) , E ( a ) σ m E (R)) defines a bijectionfrom V ( M ) / E ( R ) onto ( R/ a ) × × ( E ( a ) \ SL ( R ) / E ( R )) . ( ii ) If G is a normal subgroup of SL ( R ) containing E ( R ) , then V ( M ) /G is equipotent with ( R/ a ) × × SL ( R ) /G . In general E ( R ) is not a normal subgroup of SL ( R ) , see e.g., [Nic11, The-orem 1.5] or [Lam06, Claim I.8.16]. More tractable subgroups G could be con-sidered instead in Lemma 4.4. ii , e.g., the normal closure of E ( R ) in SL ( R ) or the group of × unipotent matrices over R . Example 4.5.
Let R be the ring of integers of a totally imaginary quadraticfield. If R is not a GE -ring, then it follows from [FF88, Theorem 2.4] and [LS77, Theorem IV.2.6] that E ( R ) \ SL ( R ) / E ( R ) is infinite (alternativelyuse [Nic11, Theorem 1.5] and [She16] ). In particular, V ( R/ a × R ) / SL ( R ) is infinite for every ideal a ⊆ R . Note that [Sta18, Proof of Proposition 7.5] offers a geometric insight on SL ( R ) / E ( R ) .Proof of Lemma 4.4. ( i ) . Given m ∈ V ( M ) , the existence of σ m is providedby Lemma 4.2. In order to see that ∆ e is well-defined, observe that det e isE ( R ) -invariant and that the stabilizer of m = ( δe , e ) in SL ( R ) is E ( a ) for every δ ∈ ( R/ a ) × . Checking that ∆ e is bijection is straightforward. ( ii ) .Check that m G (det e ( m ) , σ m G ) is well-defined and argue as in ( i ) . (cid:3) Lemma 4.6.
Let I be a two-generated ideal of R . Let a be an ideal of R suchthat glr R ( R/ a ) = 1 . Then R/ a × I is two-generated if and only if there is ( a, b ) ∈ V ( I ) such that (1 , a ) and (0 , b ) generate R/ a × I . Moreover, for every ( a, b ) ∈ V ( I ) , the following are equivalent: ( i ) R/ a × I is generated by (1 , a ) and (0 , b ) . ( ii ) ( b : a ) + a = R .If in addition we suppose I invertible, then each of the above assertions isequivalent to: ( iii ) I = I a + Rb .Proof. Assuming that µ ( R/ a × I ) ≤ , we pick an arbitrary m ∈ V ( R/ a × I ) .By hypothesis, we can find σ ∈ SL ( R ) such that Mat( m ) σ = (cid:18) a b (cid:19) forsome ( a, b ) ∈ V ( I ) , which proves the first part of the statement. ( i ) ⇒ ( ii ) . By hypothesis, there is ( λ, µ ) ∈ R such that λ (1 , a ) + µ (0 , b ) =(0 , a ) , which implies λ − ∈ ( b : a ) and λ ∈ a . ( ii ) ⇒ ( i ) . By hypothesis,there is ( a, b ) ∈ V ( I ) , λ, ν ∈ R and µ ∈ a such that λ + µ = 1 and λa = νb .As (0 , a ) = µ (1 , a ) + ν (0 , b ) , we deduce that (1 , a ) and (0 , b ) generate R/ a × I .To complete the proof, it suffices to show ( ii ) ⇔ ( iii ) under the additionalhypothesis. Multiplying both sides of ( ii ) by I , we obtain I = I a + I ( b : a ) .Clearly ( b : a ) = ( b : I ) . As I is assumed to be invertible, we have ( b : I ) = bI − , hence I = I a + Rb . Conversely, multiplying both sides of ( iii ) by I − yields R = a + ( b : a ) . (cid:3) QUIVALENT GENERATING VECTORS OF FINITELY GENERATED MODULES OVER COMMUTATIVE RINGS19
Lemma 4.7.
Let I, a be two ideals of R with I invertible and µ ( I ) ≤ . Let M + R/ a × I ⊆ R/ a × R . Then Fitt ( M ) = a .Proof. Let us consider two presentations F ϕ −→ R → R/ a → F ϕ −→ R → I → . where F and F are free R -modules. This gives rise to a presentation F ϕ −→ R → M → . with F = F × F and ϕ = ϕ × ϕ . By [Eis95, Corollary 20.4], we have Fitt ( M ) = I ( ϕ ) = I ( ϕ ) I ( ϕ ) + I ( ϕ ) = Fitt ( I ) a + Fitt ( I ) . By [Eis95,Propositions 20.5 and 20.6], we have Fitt ( I ) = { } and Fitt ( I ) = R . There-fore Fitt ( M ) = a . (cid:3) Proposition 4.8.
Let a = 0 be an ideal of R such that glr R ( R/ a ) = 1 . Let I be an invertible ideal of R and let M + R/ a × I ⊆ R/ a × R . We supposemoreover that at least one of the following holds: ( i ) I is / -generated. ( ii ) a is regular and I/Rr is cyclic for every regular element r ∈ I .Then there is ( a, b ) ∈ V ( I ) with a ∈ I a \ { } such that R/ a × I is generatedby ( e + ae , be ) where e = (1 , and e = (0 , . If m is a generating pairof this form and m ′ ∈ V ( M ) , then m ′ ∼ SL ( R ) (det m ( m ′ ) e + ae , be ) . In particular, the map det m induces a bijection from V ( M ) / SL ( R ) onto ( R/ a ) × .Proof. If ( i ) holds, we can pick a ∈ I a \ { } such that I = Ra + Rb for some b ∈ I . If ( ii ) holds, we can find two regular elements x ∈ a , y ∈ I . Setting a + xy , we also obtain that I = Ra + Rb for some b ∈ I . In particular, wehave I = I a + Rb so that m + ( e + ae , be ) generates M by Lemma 4.6.Let m ′ ∈ V ( M ) and let δ + det m ′ ( m ) . As m generates M , so does m ′′ + ( δe + ae , be ) . We claim that det m ( m ′′ ) = δ . The equivalence m ′ ∼ SL ( R ) m ′′ follows from our claim. Indeed, we have M p ≃ R p / a R p × R p for every maximalideal p of R because I is invertible, so that Proposition 4.3 applies. In orderto establish our claim, we consider the natural map π : M ։ M/ Fitt ( M ) M .Since a ∈ a = Fitt ( M ) by Lemma 4.7, we have π ( m ′′ ) = π ( m ) (cid:18) δ
00 1 (cid:19) andLemma 2.3 yields det m ( m ′′ ) = δ . This concludes the proof. (cid:3) Proposition 4.9.
Let I, a and M be as in Proposition 4.8. Assume moreoverthat er R ( R/ a ) = 1 and that at least one of the following holds: ( i ) I is / -generated and er R ( R/ b ) = 1 for every non-zero b ⊆ I . ( ii ) R is a PP-ring, a is regular, I/ b is cyclic and er R ( R/ b ) = 1 for everyregular ideal b ⊆ I .Then er ( M ) = 2 .Proof. We assume first that ( i ) holds and let n > , m ∈ V n ( M ) . We beginby proving that m ∼ E n ( R ) ( e + ae , be , , . . . , for some ( a, b ) ∈ V ( I ) with a ∈ I a \ { } . Since er R ( R/ a ) = 1 , we deduce that(4) Mat( m E ) = (cid:18) a b c (cid:19) for some E ∈ SL ( R ) , some ( a, b ) ∈ I with a = 0 and some c = ( c , . . . , c n − ) ∈ I n − . Since m E generates M , there is a row vector r = ( r, s, t ) ∈ R n with t ∈ R n − such that Mat( m E ) r ⊤ = (cid:18) a (cid:19) . As a result, we can write a = ar + bs + ct ⊤ with r ∈ a . Multiplying m E on the right by an ele-mentary matrix if needed, we can therefore assume that a ∈ I a . Since I is / -generated, we have µ ( I/Ra ) = 1 and hence er R ( I/Ra ) = 1 , so that ( b + Ra, c + Ra, . . . , c n − + Ra ) ∼ E n − ( R ) ( d + Ra, , . . . , for some d ∈ I .Thus we can assume, without loss of generality, that c i ∈ Ra for every i ≥ and I = I a + Rb . This shows in particular that ( a, b ) ∈ V ( I ) and we inferfrom Lemma 4.6 that ( e + ae , be ) generates M . It follows immediately that m ∼ E n ( R ) ( e + ae , be , , . . . , , as claimed.Let m ′ = ( e + a ′ e , b ′ e , , . . . , with ( a ′ , b ′ ) ∈ V ( I ) and a ′ ∈ I a \ { } . Weshall conclude our proof by showing that m ′ ∼ E n ( R ) m . We begin by observingthat Mat( m ′ ) ∼ E n ( R ) (cid:18) a ′ b ′ a − a ′ (cid:19) ∼ E n ( R ) (cid:18) a b ′ a − a ′ (cid:19) . Using again the fact that er R ( I/Ra ) = 1 , we deduce that ( b ′ , a − a ′ ) ∼ E ( R ) ( b + λa, µa ) for some ( λ, µ ) ∈ R . Since ( e + ae , ( b + λa ) e ) generates M byLemma 4.6, we have Mat( m ′ ) ∼ E n ( R ) (cid:18) a b + λa µa (cid:19) ∼ E n ( R ) (cid:18) a b + λa a (cid:19) from which m ′ ∼ E n ( R ) m easily follows.Let us assume now that ( ii ) holds. Proceeding as before, we only need toshow that given the identity (4) with a ∈ I a , we can further suppose that a is regular. Since R is a PP-ring, there is an idempotent e ∈ R such that Re = (0 : a ) . Let r be a regular element in I a . Then there are λ ∈ R, µ ∈ R n − such that re = λbe + µ c ⊤ e . As a + re is regular, we can therefore assume,without loss of generality, that so is a . From there on, the proof is the same. (cid:3) QUIVALENT GENERATING VECTORS OF FINITELY GENERATED MODULES OVER COMMUTATIVE RINGS21
Products of more than two factors.
In this section, we consider R -modules M of one of the two following forms R/ a × · · · × R/ a k ( k ≥ , (5) R/I × · · · × R/I k × I × R l ( k ≥ , l ≥ (6)where R ) a ⊇ · · · ⊇ a k is a descending chain of ideals, R ) I ⊇ · · · ⊇ I k is a descending chain of invertible ideals, I is either { } or an invertible idealand l > if I = { } . We show that the conclusions of Theorems A and E holdfor such modules with less restrictive assumptions on R . This is achieved withProposition 4.10 ( M as in (6), torsion free but not free), Propositions 4.12 and4.14 ( M as in (5)), Proposition 4.16 ( M as in (6) and k ≥ ) and Proposition4.18 ( M is free) below.We denote by SL k ( R ) ∩ E n ( R ) ( k ≤ n ) the subgroup of SL k ( R ) consisting ofthe matrices σ such that (cid:18) σ n − k (cid:19) ∈ E n ( R ) . Proposition 4.10.
Assume that R is Hermite. Let I be an ideal of R suchthat µ ( I ) = 2 . Let M = I × R k ⊂ R k +1 ( k ≥ . Then the following hold. ( i ) µ ( M ) = k + 2 and Fitt k +1 ( M ) = Fitt ( I ) . ( ii ) The map ( a, b ) ( ae , be , e , . . . , e k +1 ) induces a bijection from V ( I ) / SL ( R ) onto V k +2 ( M ) /G with G = SL k +2 ( R ) . If er ( R ) ≤ , wecan take G = E µ ( R ) and the above map also induces a bijection from V ( I ) / ( SL ( R ) ∩ E k +2 ( R )) onto V k +2 ( M ) / E k +2 ( R ) . ( iii ) For every n > k + 2 , the map ( a , . . . , a n − k ) ( a e , . . . , a n − k e n − k , e n − k +1 , . . . , e n ) induces a bijection from V n − k ( I ) / ( SL n − k ( R ) ∩ G ) onto V n ( M ) /G where G = SL n ( R ) . If er ( R ) ≤ , we can take G = E n ( R ) . ( iv ) If R and I are as in Proposition 3.4 or as in Proposition 3.8, then er ( M ) = k + 2 .Proof. We begin by proving ( i ) and ( ii ) simultaneously under the unique as-sumption glr ( R ) = 1 . Let m ∈ V µ ( M ) . First, we shall prove by inductionon k ≥ that µ = k + 2 and that m ∼ SL µ ( R ) ( ae , be , e , . . . , e k +1 ) for some ( a, b ) ∈ V ( I ) depending on m , which is obvious for k = 0 . Suppose now that k ≥ . As glr ( R ) = 1 , we deduce that(7) Mat( m ) ∼ SL µ ( R ) (cid:18) A b0 (cid:19) with A a k × ( µ − matrix and b a column vector of size k . It follows fromLemma 4.1 that (cid:18) A b0 (cid:19) ∼ E µ ( R ) (cid:18) A (cid:19) . Our claim is then proven by applying the induction hypothesis to the gen-erating vector m ′ ∈ V µ − ( M ′ ) defined by Mat( m ′ ) = A .The identity Fitt k +1 ( M ) = Fitt ( I ) follows from Lemma 2.2 applied to m =( ae , be , e , . . . , e k +1 ) . We have already established that the map ( a, b ) ( ae , be , e , . . . , e k +1 ) induces a surjection from V ( I ) / SL ( R ) onto V k +2 ( M ) / SL k +2 ( R ) . In orderto show that the induced map is also injective, let us consider ( a, b ) , ( a ′ , b ′ ) ∈ V ( I ) and σ ∈ SL k +2 ( R ) such that ( a ′ e , b ′ e , e , e , . . . , e k +1 ) = ( ae , be , e , e , . . . , e k +1 ) σ. Then σ must be of the form (cid:18) σ ′ k − (cid:19) with σ ′ ∈ SL ( R ) . Thus ( a ′ , b ′ ) ∼ SL ( R ) ( a, b ) .If in addition er ( R ) ≤ , we show likewise that m ∼ E k +2 ( R ) ( ae , be , e , . . . , e k +1 ) for some ( a, b ) ∈ V ( I ) by induction on k . Since er ( R ) ≤ < k + 2 , we canuse E k +2 ( R ) to reduce Mat( m ) in (7). From there, the proof follows the samelines. ( iii ) . Proceed by induction on k as in the proof of assertion ( ii ) .Assertion ( iv ) follows from ( iii ) , since E n − k ( R ) acts then transitively on V n − k ( I ) by Propositions 3.4 and 3.8. (cid:3) Lemma 4.11.
Let M = R/ a ×· · ·× R/ a k ( k ≥ where the ideals a i ⊆ R forman descending chain R ) a ⊇ · · · ⊇ a k . Then µ ( M ) = k and Fitt k − ( M ) = a .Proof. Clearly µ ( M ) ≤ k . Let p be a maximal ideal of R containing a . Then M/ p M surjects onto ( R/ p ) k which is vector space of dimension k over R/ p .Therefore µ ( M ) ≥ k and hence µ ( M ) = k . The identity Fitt k − ( M ) = a follows from Lemma 2.2 applied to m = ( e , . . . , e k ) . (cid:3) Proposition 4.12.
Let M be as in Lemma 4.11. Let G ∈ { SL n ( R ) , E n ( R ) } with n ≥ k and let e = ( e , . . . , e k ) . Assume that one of the following holds: ( i ) G = SL n ( R ) and glr R ( R/ a i ) ≤ i for every i ≥ . ( ii ) G = E n ( R ) and er R ( R/ a i ) ≤ max(1 , i − for every i ≥ .Then for every m ∈ V n ( M ) , we have: m ∼ G (cid:26) (det e ( m ) e , e , . . . , e k ) , if n = k, ( e , e , . . . , e k , , . . . , , if n > k. QUIVALENT GENERATING VECTORS OF FINITELY GENERATED MODULES OVER COMMUTATIVE RINGS23
In particular det e induces a bijection from V k ( M ) /G onto ( R/ a ) × . Remark 4.13.
The hypothesis ( i ) of Proposition 4.12 is clearly satisfied if R is a K-Hermite ring. It is also satisfied if at least one of the following holds: ( a ) Every proper quotient of R has stable rank . ( b ) The ring R is as in Proposition 3.8 and each ideal a i is either { } orregular.Hypothesis ( ii ) is satisfied if any of ( a ) or ( b ) holds and a = { } .Proof of Proposition 4.12. We shall proceed by induction on k ≥ . If k = 1 ,the result holds trivially under any of the assumptions. We assume throughoutthat k ≥ and hypothesis ( ii ) holds, in particular G = E n ( R ) . The proof canbe trivially adapted to hypothesis ( i ) .Consider m ∈ V n ( M ) with n ≥ k . Since n > er R ( R/ a k ) , we have Mat( m ) ∼ E n ( R ) (cid:18) A b0 (cid:19) with A a k × ( n − matrix and b a column vector of size k . By Lemma 4.1,we obtain Mat( m ) ∼ E n ( R ) (cid:18) A (cid:19) . If n = k , we use the induction hypothesis for the generating vector m ′ ∈ V k − ( M ′ ) satisfying A = Mat( m ′ ) to infer that m ∼ E k ( R ) ( δe , e , . . . , e k ) forsome δ ∈ ( R/ a ) × . It follows from Lemmas 2.2 and 4.11 that δ = det e ( m ) . If n > k , we infer that m ∼ E n ( R ) ( e , . . . , e k − , , . . . , , e k ) and using one furtherelementary matrix, we obtain m ∼ E n ( R ) ( e , e , . . . , e k , , . . . , . (cid:3) Proposition 4.14.
Let M be as in Lemma 4.11. Let d ≥ and n > k + d .Assume that er R ( R/ a i ) ≤ i + d for every i ≥ . Then for every m ∈ V n ( M ) ,we have m ∼ E n ( e , e , . . . , e k , , . . . , .Proof. Proceed by induction on k ≥ as in the proof of Proposition 4.12. (cid:3) Lemma 4.15.
Let M be a finitely generated R -module and let a , . . . , a k beideals of R such that a + · · · + a k ⊆ ann( M ) . Let m ∈ V n ( M × R/ a × · · · × R/ a k ) with n ≥ µ ( M × R/ a × · · · × R/ a k ) be such that Mat( m ) = (cid:18) m m k (cid:19) where m ∈ M n − k and m ∈ M k . Then every component of m is an R -linearcombination of the components of m . In particular, we have Mat( m ) ∼ E n ( R ) (cid:18) m k (cid:19) . Proof.
Let us write m = ( m , . . . , m n − k ) , m = ( m n − k +1 , . . . , m n ) and fix acomponent m of m . Since m generates M × R/ a × · · · × R/ a k , there is r =( r , . . . , r n ) ∈ R n such that Mat( m ) r ⊤ = (cid:18) m (cid:19) . Hence we have m = P ni =1 r i m i and r n − k + i ∈ a i for ≤ i ≤ k . Since a i ⊆ ann( M ) for every i by hypothesis,we actually have m = P n − ki =1 r i m i . (cid:3) Proposition 4.16.
Let I be a non-principal invertible ideal of R .Let M = R/ a × · · · × R/ a k × I × R l ( k ≥ , l ≥ where the ideals a i ⊆ R form an descending chain R ) a ⊇ · · · ⊇ a k and a k = 0 .Assume moreover that at least one of the following holds: ( a ) Every proper quotient of R has stable rank and I is / -generated. ( b ) R and I are as in Proposition 3.8 and each ideal a i is regular.Let e + ( e , . . . , e k + l +1 ) , ( a, b ) ∈ V ( I ) with a ∈ I a k .Then the following hold: ( i ) µ ( M ) = k + l + 1 and Fitt k + l ( M ) = a . ( ii ) For every m ∈ V k + l +1 ( M ) , we have m ∼ G ( det e ( m ) e , e , . . . , e k , e k + ae k +1 , be k +1 , e k +2 , . . . , e k + l +1 ) for G = SL k + l +1 ( R ) . This holds also for G = E k + l +1 ( R ) if k > . ( iii ) For every m ∈ V n ( M ) , we have m ∼ E n ( R ) ( e , e , . . . , e k , e k + ae k +1 , be k +1 , e k +2 , . . . , e k + l +1 , , . . . , if n > k + l + 1 .In particular, det e induces a bijection from V k + l +1 ( M ) / SL k + l +1 ( R ) onto ( R/ a ) × .Proof. ( i ) . Let p be a maximal ideal of R containing a . Then M surjectsonto ( R/ p ) k × I/I p × ( R/ p ) l . By hypothesis I/I p is cyclic so that I/I p ≃ R/ p by Lemma 3.2. Therefore M surjects onto ( R/ p ) k + l +1 and hence cannot begenerated by less than k + l + 1 elements. Since R/ a k × I can be gener-ated by two elements thanks to Proposition 4.8, M can be generated by k + l + 1 elements. Thus µ ( M ) = k + l + 1 . Let m = ( e , e , . . . , e k , e k + ae k +1 , be k +1 , e k +2 , . . . , e k + l +1 ) . Then m generates M and it follows from Lemma2.2 that Fitt k + l ( M ) = a + Fitt ( R/ a k × I ) . Since Fitt ( R/ a k × I ) = a k byLemma 4.7, we obtain that Fitt k + l ( M ) = a . ( ii ) . We shall prove by induction on l ≥ that m ∼ G ( δe , e , . . . , e k , e k + ae k +1 , be k +1 , e k +2 , . . . , e k + l +1 ) QUIVALENT GENERATING VECTORS OF FINITELY GENERATED MODULES OVER COMMUTATIVE RINGS25 for some δ = δ ( m ) ∈ ( R/ a ) × . The identity δ = det e ( m ) is a straightforwardconsequence of assertion ( i ) and Lemma 2.2.If l = 0 and k = 1 , assertion ( ii ) is given by Proposition 4.8.If l = 0 and k > , Proposition 4.9 yields Mat( m ) ∼ E k + l +1 ( R ) A b c0 a b where A is a ( k + l − × ( n − matrix, b and c are column vectors of size k + l − . By Lemma 3.2, the map b induces an isomorphism from I/I a k onto R/ a k . This isomorphism allows us to identify M/ a k M with R/ a × · · · × R/ a k × R/ a k . Under this identification the image m of m in ( M/ a k M ) k + l +1 satisfies Mat( m ) = A b c0 . By Lemma 4.15, the vectors b and c are inthe R -linear span of the columns of A . Thus Mat( m ) ∼ E k + l +1 ( R ) A a b .We complete the proof of the induction basis by applying Proposition 4.12 tothe generating vector m ′ ∈ V k ( R/ a × · · · × R/ a k ) defined by Mat( m ′ ) = A .Assume now that l > . We have er ( R ) ≤ by Lemma 3.3 under assumption ( a ) and by Proposition 3.8 under assumption ( b ) . Thus Mat( m ) ∼ E k + l +1 ( R ) (cid:18) A b0 (cid:19) ∼ E k + l +1 ( R ) (cid:18) A (cid:19) where A is a ( k + l ) × ( n − matrix, b a column vector of size k + l and thesecond equivalence is given by Lemma 4.1. Our claim follows now from theinduction hypothesis applied to the generating vector m ′ ∈ V k + l ( R/ a × · · · × R/ a k × I × R l − ) defined by Mat( m ′ ) = A . ( iii ) . We proceed again by induction on l ≥ . Assume that l = 0 . Ifmoreover k = 1 , then assertion ( iii ) is given by Proposition 4.9. If k > ,reduce Mat( m ) as in ( ii ) and use Proposition 4.12 to complete the proof ofthis induction basis. If l > , reduce Mat( m ) as in ( ii ) and apply the inductionhypothesis. (cid:3) Proposition 4.17.
Let R be a Hermite ring. Then the following hold. V ( R ) / E ( R ) ≃ E ( R ) \ SL ( R ) / E ( R ) , (8) V n ( R ) / E n ( R ) ≃ SL n − ( R ) E n ( R ) \ SL n ( R ) for n > , (9) V n ( R n ) / E n ( R ) ≃ R × × ( SL n ( R ) / E n ( R )) for n ≥ , (10) V n ( R l ) / E n ( R ) ≃ SL n − l ( R ) E n ( R ) \ SL n ( R ) for n > l ≥ . (11) Proof. (8) Apply Lemma 4.4 with a = R . (9) We have V n ( R ) = e SL n ( R ) .The stabilizer of e for the action of SL n ( R ) is SL n − ( R ) . As E n ( R ) is normal inSL n ( R ) by Suslin’s Normality Theorem [Mag02, Theorem 10.8], the result fol-lows. (10) By Proposition 4.12, we have V n ( R ) = F u ∈ R × ( ue , e , . . . , e n ) SL n ( R ) .As the stabilizer of ( ue , e , . . . , e n ) for the action of SL n ( R ) is trivial, the resultfollows. (11) By Proposition 4.12, we have V n ( R ) = ( e , e , . . . , e l , , . . . , SL n ( R ) . The stabilizer S of ( e , e , . . . , e l , , . . . , for the action of SL n ( R ) satisfiesSL n − l ( R ) ⊂ S ⊂ E n ( R ) SL n − l ( R ) . Since E n ( R ) is normal in SL n ( R ) , the resultfollows. (cid:3) Proposition 4.18.
Let R be a Hermite ring. ( i ) If n > l + sr ( R ) − , then E n ( R ) acts transitively on V n ( R l ) . ( ii ) If sr ( R ) ≤ , then we have V n ( R n ) / E n ( R ) ≃ R × × SK ( R ) for n > , (12) V n +1 ( R n ) / E n +1 ( R ) ≃ SK ( R ) for n ≥ . (13) Proof. ( i ) . This is an immediate consequence of Proposition 4.14. ( ii ) . Com-bine the identifications (10) and (11) of Proposition 4.17 with SK StabilityTheorem [Mag02, Corollary 11.19. v ]. (cid:3) Lemma 4.19.
Let a be an ideal of R and let M be a finitely generated R -module. Assume that at least one of the following holds. ( i ) R is Hermite and M = R/ a × R l with l > . ( ii ) R and I are as in Proposition 3.8, a is regular and M = R/ a × I × R l with l > .Then a = Fitt µ − ( M ) and V µ ( M ) / E µ ( R ) is equipotent with ( R/ a ) × × SL µ ( R ) / E µ ( R ) where µ = µ ( M ) . If in addition sr ( R ) ≤ , e.g., if ( ii ) holds, then V µ ( M ) / E µ ( R ) can be further identified with ( R/ a ) × × SK ( R ) .Proof. The identity a = Fitt µ − ( M ) is given by Lemma 4.11 and Proposition4.16. If ( i ) , respectively ( ii ) holds, then V µ ( M ) = F u ∈ ( R/ a ) × m u SL µ ( R ) where m u = ( ue , e , . . . , e µ ) , respectively m u = ( ue , e + ae , be , e , . . . , e µ ) and µ = l , respectively µ = l + 2 , by Propositions 4.12 and 4.16. In each case, itis easily checked that the stabilizer S u of m u for the action of SL µ ( R ) consistsin the matrices of the form (cid:18) r0 µ − (cid:19) QUIVALENT GENERATING VECTORS OF FINITELY GENERATED MODULES OVER COMMUTATIVE RINGS27 where r is a row vector of length µ − whose components lie in a . Therefore S u ⊂ E µ ( R ) . Since E µ ( R ) is normal in SL µ ( R ) by Suslin’s Normality Theo-rem [Mag02, Theorem 10.8], we deduce that V µ ( M ) / E µ ( R ) is equipotent with ( R/ a ) × × SL µ ( R ) / E µ ( R ) . If sr ( R ) ≤ , then SL µ ( R ) / E µ ( R ) ≃ SK ( R ) by the SK Stability Theorem [Mag02, Corollary 11.19. v ]. Hence V µ ( M ) / E µ ( R ) ≃ ( R/ a ) × × SK ( R ) . (cid:3) Direct products of rings and quotient rings
The proofs of Theorems A and E rely on the existence of invariant decom-positions defined in Theorems 2.9 and 2.12. The results in Section 4.2 don’taddress the decomposition of a module M as in Theorem 2.12 when non-trivialidempotents e i split the torsion-free part of M . They are however sufficientto prove Theorems A and E in full generality. Indeed, thanks to the splittings R ≃ Q i Re i and M ≃ Q i e i M , Proposition 5.3 below allows us to handle thesubmodules e i M independently. We also prove Proposition C by establishingPropositions 5.3 and 5.4 below.Given a ring R , let E ( R ) + S n E n ( R ) and SL ( R ) + S n SL n ( R ) be theascending unions for which the embeddings E n ( R ) → E n +1 ( R ) and SL n ( R ) → SL n +1 ( R ) are defined through A (cid:18) A
00 1 (cid:19) . The special Whitehead group SK ( R ) of R is the quotient group SL ( R ) / E ( R ) . Lemma 5.1.
Let { R x } be a family of rings where x ranges in a set X and let R = Q x R x . Then the following hold. ( i ) The natural maps SL n ( R ) → Q x SL n ( R x ) and GL n ( R ) → Q x GL n ( R x ) are isomorphisms for every n ≥ . ( ii ) If X is finite, then the natural map E n ( R ) → Q x E n ( R x ) is an isomor-phism for every n ≥ . ( iii ) Let n ≥ . If every matrix in E n ( R x ) is the product of at most ν n ele-mentary matrices where ν n < ∞ is independent on x , then the naturalmap E n ( R ) → Q x E n ( R x ) is an isomorphism. ( iv ) If X is finite, or if sr ( R x ) is bounded and the natural map E n ( R ) → Q x E n ( R x ) is an isomorphism, then the natural map SK ( R ) → Y x SK ( R x ) is an isomorphism.Proof. We first observe that the natural maps considered in the assertions ( i ) to ( iii ) are clearly injective. Assertion ( i ) is obvious. For ( ii ) , see [Coh66,Proof of Theorem 3.1] or reason as in ( iii ) . ( iii ) . Let us prove that the natural map ϕ : E n ( R ) → Q x E n ( R x ) issurjective. Let ( E x ) ∈ Q x E n ( R x ) . Write E x = E x, · · · E x,ν n with E x,k = E i ( x ) j ( x ) ( r x,k ) ( r x,k ∈ R x ) where E ij ( r ) denotes the elementary matrix whose ( i, j ) -coefficient is r and whose other off-diagonal coefficients are zero. Thenwe have E x = Q ν n k =1 Q ≤ i = j ≤ n E ij ( r x,k ( i, j )) where the factors indexed by thepairs ( i, j ) are ordered lexicographically and r x,k ( i, j ) = (cid:26) r x,k if ( i, j ) = ( i ( x ) , j ( x )) , else.Set E + Q ν n k =1 Q ≤ i = j ≤ n E ij ( r kij ) , with r kij = ( r k,x ( i, j )) ∈ R . Then ϕ ( E ) =( γ x ) , which proves that ϕ is an isomorphism. ( iv ) . It is trivial to check that the natural map ψ : SK ( R ) → Q x SK ( R x ) isalways surjective. In order to show that ψ is injective, consider σ E ( R ) ∈ ker( ψ ) with σ = ( σ x ) ∈ SL n ( R ) for some n ≥ . Then σ x ∈ E ( R x ) for every x ∈ X . If X is finite, then we can find N ≥ , such σ x ∈ E N ( R x ) for every x . Therefore ( σ x ) ∈ Q x E N ( R x ) and the result follows from ( iii ) . If sr ( R x ) < N for some N ≥ , then σ x ∈ E N ( R x ) for every x by the SK Stability Theorem [Mag02,Corollary 11.19. v ]. Thus ( σ x ) ∈ E ( R ) , provided that E N ( R ) → Q x E N ( R x ) isan isomorphism. (cid:3) Lemma 5.2.
Let { R x } be a family of rings where x ranges in a set X . Let M be module over R + Q x R x . Denote by e x the element of R whose componentin R x is and whose other components are zero. Let ϕ : M → Q x e x M be the R -homomorphism defined by ϕ ( m ) = ( e x m ) . Then the following hold. ( i ) If X is finite, then ϕ is an isomorphism. ( ii ) If M is finitely generated, then ϕ is surjective. ( iii ) If M is finitely presented, then ϕ is an isomorphism.Proof. ( i ) . The identity of R decomposes as P x e x . Thus, for every m ∈ M , we have m = 1 · m = P x e x m , which shows that ϕ is injective. If ( m x ) ∈ Q x e x M , we can set m + P x e x m x so that ϕ ( m ) = ( m x ) . Hence ϕ issurjective. ( ii ) . Let ( m , . . . , m n ) be a generating vector of M . Let ( m x ) ∈ Q x e x M .Since ( e x m , . . . , e x m n ) generates e x M , we can find λ i,x ∈ R x (1 ≤ i ≤ n ) such that m x = P i λ i,x e x m i . Setting λ i + ( λ i,x ) x ∈ R , we observe that ( m x ) = ϕ ( P i λ i m i ) . Therefore ϕ is surjective. ( iii ) . By hypothesis M is the cokernel of an n × m matrix A associated toa generating vector ( m , . . . , m n ) ∈ M n . Let c , . . . , c m ∈ R n be the columnsof A . Given λ = ( λ , . . . , λ n ) ∈ R n , the element P i λ i m i is zero if and only if λ lies in the R -linear span of the columns c j . Let m = P i λ i m i ∈ M be suchthat ϕ ( m ) = 0 . Since for every x ∈ X we have P i e x λ i m i = 0 , it follows that e x λ = P j µ j,x c j for some µ j,x ∈ R with ≤ j ≤ m . Setting µ j = ( µ j,x ) x , we QUIVALENT GENERATING VECTORS OF FINITELY GENERATED MODULES OVER COMMUTATIVE RINGS29 observe that λ = P j µ j c j . Therefore m = 0 , which shows that ϕ is injective.As ϕ is also surjective by ( ii ) , it is an isomorphism. (cid:3) Proposition 5.3.
Let { R x } be a family of rings where x ranges in a set X .Let M be a finitely presented module over R + Q x R x . ( i ) If the R x -module e x M has property ∆ SL for every x ∈ X , then M hasproperty ∆ SL . ( ii ) Assume that the natural map E n ( R ) → Q x E n ( R x ) is an isomorphismfor every n ≥ µ ( M ) . If in addition the R x -module e x M has property ∆ E for every x ∈ X , then M has property ∆ E .Proof. We shall only consider the property ∆ SL . The corresponding result for ∆ E can be derived in the same way. Assume that ( i ) holds. Let n ≥ µ ( M ) .Using the natural isomorphism SL µ ( R ) ≃ Q x SL µ ( R x ) and Lemma 5.2. iii , weobtain a bijection V n ( M ) / SL n ( R ) ≃ Q x V n ( e x M ) / SL n ( R x ) . By hypothesisthe determinant map induces a bijection V n ( e x M ) / SL n ( R x ) ≃ V ( V n e x M ) for every x . It is easily checked that Q x V ( V n e x M ) ≃ V ( V n M ) by meansof the natural R -isomorphism Q x V n e x M ≃ V n M . The result follows. (cid:3) Proposition 5.4.
Let R be a ring and let R be a quotient of R . ( i ) If R satisfies ∆ E , then R satisfies ∆ E . ( ii ) If R satisfies ∆ SL and if the natural map SL n ( R ) → SL n ( R ) is surjectivefor every n , then R satisfies ∆ SL . Remark 5.5.
The natural map SL n ( R ) → SL n ( R ) is surjective for every n and every quotient R of R if sr ( R ) ≤ [EO67, Corollary 8.3] . See also thepartial stability result [EO67, Theorem 8.3] when R is a ring of univariatepolynomials over a principal ideal domain.Proof. ( i ) . The quotient map R ։ R endows R with an R -algebra structure.Let M be an R -module and let M be the R -module with underlying set M thatis obtained from M by restricting scalars. We have the following commutativediagram V n ( M ) / E n ( R ) V ( V n M )V n ( M ) / E n ( R ) V ( V n M ) detdet where the left vertical arrow is given by the surjectivity of the natural mapE n ( R ) → E n ( R ) and the right vertical arrow is given by the base changeisomorphism [Eis95, Proposition A2.2.b]. The bottom horizontal arrow is abijection as the other arrows are bijections. ( ii ) . Proceed as in ( i ) . (cid:3) Proofs
Proof of Theorem A. If R is an elementary divisor ring, then R is K-Hermiteand M has an invariant decomposition in the sense of Theorem 2.9. The resultfollows by combining Propositions 4.10 and 4.12.Let us suppose that R is an almost-LG coherent Prüfer ring. Let ( e i ) i be theidempotents of R appearing in the invariant decomposition of M in Theorem2.12. Then each Re i is an almost-LG coherent Prüfer ring and M is isomorphicto Q i e i M by Lemma 5.2. i . By Proposition 5.3. i , we can assume, without lossof generality, that M is of the form R/I × · · · × R/I k × I × R l ( k ≥ , l ≥ where R ) I ⊇ · · · ⊇ I k is a descending chain of invertible ideals and I iseither { } or an invertible ideal. As R is a coherent Prüfer ring, it is certainlya PP-ring. If I is invertible, then R and I satisfy the hypotheses of Proposition3.8 by [Cou07, Lemma 10]. In particular er ( R ) ≤ and R is Hermite. Theresult follows by combining Propositions 4.10, 4.12 and 4.16. (cid:3) Proof of Theorem E.
Let us assume first that R is a Bézout ring whoseproper quotients have stable rank . By [McG08, Theorem 3.7], the ring R isan elementary divisor ring of stable rank at most . ( i ) . If µ = 1 , the result is trivial. Let us suppose that µ ′ > . By Theorem2.9, the module M is of the form R/ a ×· · ·× R/ a k ( k > where a ⊇ · · · ⊇ a k is a descending chain of ideals such that R ) a and a = { } . Then the resultfollows from Proposition 4.12 and Remark 4.13. ( ii ) . As M is of the form R/ a × R l ( l > with R ) a = { } , the resultfollows from Lemma 4.19. ( iii ) . As sr ( R ) ≤ , Proposition 4.14 applies with d = 0 if µ ′ > . Otherwise,there is nothing to show since R is a Bézout ring. ( iv ) . Proposition 4.14 applies with d = 1 .Let us assume now that R is an almost-LG coherent Prüfer ring. ( i ) . If µ = 1 , the result is trivial. Let us suppose that µ ′ > . Because ofTheorem 2.12, Proposition 5.3. ii and Lemma 5.1. ii , we can assume, withoutloss of generality, that M is of the form R/I ×· · · × R/I k × I × R l ( k > , l ≥ where R ) I ⊇ · · · ⊇ I k = { } is a descending chain of invertible ideals and I is either { } or an invertible ideal. Then the result follows from Proposition4.12, Remark 4.13 and Proposition 4.16. ii ( ii ) . We can assume, without loss of generality, that M is of the form R/I × I × R l ( l > where I is an invertible ideal, I is either invertible or { } and l > if I = { } . Then the result follows from Lemma 4.19. ( iii ) . Let us assume first that µ ′ > . We can assume without loss ofgenerality that M is of the form R/I × · · · × R/I k × I × R l ( k > , l ≥ QUIVALENT GENERATING VECTORS OF FINITELY GENERATED MODULES OVER COMMUTATIVE RINGS31 where R ) I ⊇ · · · ⊇ I k = { } is a descending chain of invertible ideals and I is either { } or an invertible ideal. If µ = 1 , then M = R/I . As sr ( R/I ) = 1 ,Proposition 2.7. iii yields the result. If µ > then the result follows fromProposition 4.9, Proposition 4.12 and Proposition 4.16. iii .Suppose now that µ ′ = 0 and that eM surjects onto an invertible non-principal ideal of Re where e is an idempotent of R satisfying µ ( eM ) = µ ( M ) .We can assume, without loss of generality, that e is the last idempotent inthe decomposition of Theorem 2.12. We can also assume that M is of theform I × R l ( l ≥ with I invertible, in which case the result is given byProposition 3.8 and Proposition 4.10. iv . Indeed, if e ′ is an idempotent elementof R such that µ ( e ′ M ) < µ ( M ) , then E n ( R ) acts transitively on V n ( e ′ M ) forevery n > µ ( M ) by Proposition 3.8, Proposition 4.10. iv and Proposition 4.18. i . ( iv ) . By ( iii ) , we can assume that M is free, so that the result follows fromProposition 4.18. i . (cid:3) References [Bas64] H. Bass. K -theory and stable algebra. Inst. Hautes Études Sci. Publ. Math. ,(22):5–60, 1964. 10[Bas75] H. Bass. Libération des modules projectifs sur certains anneaux de polynômes.In
Séminaire Bourbaki, 26e année (1973/1974), Exp. No. 448 , pages 228–354.Lecture Notes in Math., Vol. 431. 1975. 2[Che17] J. Chen. Infinite prime avoidance. Preprint, https://arxiv.org/abs/1710.05496v1,October 2017. 4[CK83] D. Carter and G. Keller. Bounded elementary generation of SL n ( O ) . Amer. J.Math. , 105(3):673–687, 1983. 4[CLQ04] T. Coquand, H. Lombardi, and C. Quitté. Generating non-Noetherian modulesconstructively.
Manuscripta Math. , 115(4):513–520, 2004. 10[Coh58] P. Cohn. Rings of zero-divisors.
Proc. Amer. Math. Soc. , 9:909–914, 1958. 3[Coh66] P. Cohn. On the structure of the GL of a ring. Inst. Hautes Études Sci. Publ.Math. , (30):5–53, 1966. 4, 8, 27[Cou07] F. Couchot. Finitely presented modules over semihereditary rings.
Comm. Algebra ,35(9):2685–2692, 2007. 2, 10, 11, 12, 13, 30[DB72] B. Dulin and H. Butts. Composition of binary quadratic forms over integral do-mains.
Acta Arith. , 20:223–251, 1972. 11[DTZ18] M. Das, S. Tikader, and M. Zinna. Orbit spaces of unimodular rows over smoothreal affine algebras.
Invent. Math. , 212(1):133–159, 2018. 2[EG82] D. Estes and R. Guralnick. Module equivalences: local to global when primitivepolynomials represent units.
J. Algebra , 77(1):138–157, 1982. 11[Eis95] D. Eisenbud.
Commutative algebra , volume 150 of
Graduate Texts in Mathematics .Springer-Verlag, New York, 1995. With a view toward algebraic geometry. 7, 13,15, 19, 29[EO67] D. Estes and J. Ohm. Stable range in commutative rings.
J. Algebra , 7:343–362,1967. 4, 29 [Fas11] J. Fasel. Some remarks on orbit sets of unimodular rows.
Comment. Math. Helv. ,86(1):13–39, 2011. 2[FF88] C. Frohman and B. Fine. Some amalgam structures for Bianchi groups.
Proc.Amer. Math. Soc. , 102(2):221–229, 1988. 4, 18[FS01] L. Fuchs and L. Salce.
Modules over non-Noetherian domains , volume 84 of
Math-ematical Surveys and Monographs . American Mathematical Society, Providence,RI, 2001. 12[GH56] L. Gillman and M. Henriksen. Some remarks about elementary divisor rings.
Trans. Amer. Math. Soc. , 82:362–365, 1956. 11[GH70] R. Gilmer and W. Heinzer. On the number of generators of an invertible ideal.
J.Algebra , 14:139–151, 1970. 13, 15[GMV94] F. Grunewald, J. Mennicke, and L. Vaserstein. On the groups SL ( Z [ x ]) and SL ( k [ x, y ]) . Israel J. Math. , 86(1-3):157–193, 1994. 5[Guy17] L. Guyot. On finitely generated modules over quasi-euclidean rings.
Arch. Math.(Basel). , 2017. 1, 2[Guy20] L. Guyot. Equivalent generating pairs of an ideal of a commutative ring.
Preprint ,2020. 2, 4, 8, 15, 16, 17[Hei76] R.. Heitmann. Generating ideals in Prüfer domains.
Pacific J. Math. , 62(1):117–126, 1976. 13[Hei84] R. Heitmann. Generating non-Noetherian modules efficiently.
Michigan Math. J. ,31(2):167–180, 1984. 10[HSG86] J. Hermida and T. Sánchez-Giralda. Linear equations over commutative rings anddeterminantal ideals.
J. Algebra , 99(1):72–79, 1986. 11[Kap49] I. Kaplansky. Elementary divisors and modules.
Trans. Amer. Math. Soc. , 66:464–491, 1949. 11[Lam00] T. Lam. Bass’s work in ring theory and projective modules. arXiv:0002217v1[math.RA] 25 Feb 2000, February 2000. 5[Lam01] T. Lam.
A first course in noncommutative rings , volume 131 of
Graduate Textsin Mathematics . Springer-Verlag, New York, second edition, 2001. 9[Lam06] T. Lam.
Serre’s problem on projective modules . Springer Monographs in Mathe-matics. Springer-Verlag, Berlin, 2006. 2, 8, 18[LLS74] M. Larsen, W. Lewis, and T. Shores. Elementary divisor rings and finitely pre-sented modules.
Trans. Amer. Math. Soc. , 187:231–248, 1974. 11[LM88] D. Lantz and M. Martin. Strongly two-generated ideals.
Comm. Algebra ,16(9):1759–1777, 1988. 13[LQ15] H. Lombardi and C. Quitté.
Commutative algebra: constructive methods , vol-ume 20 of
Algebra and Applications . Springer, Dordrecht, revised edition, 2015.Finite projective modules, Translated from the French by Tania K. Roblot. 11,12, 15[LS77] R. Lyndon and P. Schupp.
Combinatorial group theory . Springer-Verlag, Berlin,1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. 4, 18[LW85] L. Levy and R. Wiegand. Dedekind-like behavior of rings with -generated ideals. J. Pure Appl. Algebra , 37(1):41–58, 1985. 5[Mag02] B. Magurn.
An algebraic introduction to K -theory , volume 87 of Encyclopedia ofMathematics and its Applications . Cambridge University Press, Cambridge, 2002.12, 26, 27, 28
QUIVALENT GENERATING VECTORS OF FINITELY GENERATED MODULES OVER COMMUTATIVE RINGS33 [Mat83] E. Matlis. The minimal prime spectrum of a reduced ring.
Illinois J. Math. ,27(3):353–391, 1983. 16[Mat89] H. Matsumura.
Commutative ring theory , volume 8 of
Cambridge Studies in Ad-vanced Mathematics . Cambridge University Press, Cambridge, second edition,1989. Translated from the Japanese by M. Reid. 12[McG08] W. McGovern. Bézout rings with almost stable range 1.
J. Pure Appl. Algebra ,212(2):340–348, 2008. 5, 10, 14, 30[MM82] P. Menal and J. Moncasi. On regular rings with stable range . J. Pure Appl.Algebra , 24(1):25–40, 1982. 10[MR87] J. McConnell and J. Robson.
Noncommutative Noetherian rings . Pure and AppliedMathematics (New York). John Wiley & Sons, Ltd., Chichester, 1987. With thecooperation of L. W. Small, A Wiley-Interscience Publication. 8, 9[Mur03] M. Murthy. Generators of a general ideal. In
A tribute to C. S. Seshadri (Chennai,2002) , Trends Math., pages 379–384. Birkhäuser, Basel, 2003. 2[MW81] B. McDonald and W. Waterhouse. Projective modules over rings with many units.
Proc. Amer. Math. Soc. , 83(3):455–458, 1981. 11[Nic11] B. Nica. The unreasonable slightness of E over imaginary quadratic rings. Amer.Math. Monthly , 118(5):455–462, 2011. 5, 18[Rie83] M. Rieffel. Dimension and stable rank in the K -theory of C ∗ -algebras. Proc. Lon-don Math. Soc. (3) , 46(2):301–333, 1983. 8[She16] A. Sheydvasser. A corrigendum to unreasonable slightness.
Amer. Math. Monthly ,123(5):482–485, 2016. 5, 18[Sho74] T. Shores. Modules over semihereditary Bezout rings.
Proc. Amer. Math. Soc. ,46:211–213, 1974. 3, 11[Sta18] K. Stange. Visualizing the arithmetic of imaginary quadratic fields.
Int. Math.Res. Not. IMRN , (12):3908–3938, 2018. 18[Vas72] L. Vaseršte˘ın. The group SL over Dedekind rings of arithmetic type. Mat. Sb.(N.S.) , 89(131):313–322, 351, 1972. 4[vdK77] W. van der Kallen. The K of rings with many units. Ann. Sci. École Norm. Sup.(4) , 10(4):473–515, 1977. 11[vdK83] W. van der Kallen. A group structure on certain orbit sets of unimodular rows.
J. Algebra , 82(2):363–397, 1983. 2[VS76] L. Vaseršte˘ın and A. Suslin. Serre’s problem on projective modules over polyno-mial rings, and algebraic K -theory. Izv. Akad. Nauk SSSR Ser. Mat. , 40(5):993–1054, 1199, 1976. 2[Wei13] C. Weibel.
The K -book , volume 145 of Graduate Studies in Mathematics . Amer-ican Mathematical Society, Providence, RI, 2013. An introduction to algebraic K -theory. 9 EPFL ENT CBS BBP/HBP. Campus Biotech. B1 Building, Chemin des mines,9, Geneva 1202, Switzerland
Email address ::