aa r X i v : . [ m a t h . A C ] N ov An extension of S –noetherian rings and modules P. Jara
Department of AlgebraUniversity of Granada
Abstract
For any commutative ring A we introduce a generalization of S –noetherian ringsusing a hereditary torsion theory σ instead of a multiplicatively closed subset S ⊆ A .It is proved that if A is a totally σ –noetherian ring, then σ is of finite type, and thattotally σ –noetherian is a local property. Introduction In [ ] , the authors study the problem of determining the structure of the polynomial ring D [ X ] , over an integral domain D with field of fractions K , through the structure of theEuclidean domain K [ X ] . In particular, an ideal a ⊆ D [ X ] is said to be almost principal whenever there exist a polynomial F ∈ a , of positive degree, and an element 0 = s ∈ D such that a s ⊆ F D [ X ] ⊆ a . The integral domain D is an almost principal domain whenever every ideal a ⊆ D [ X ] , which extends properly to K [ X ] , is almost principal.Noetherian and integrally closed domains are examples of almost principal domains.Later, in [ ] , the authors extend this notion to non–necessarily integral domains in defin-ing, for a given multiplicatively closed subset S ⊆ A of a ring A , an ideal a ⊆ A to be S –finite if there exist a finitely generated ideal a ′ ⊆ a and an element s ∈ S such that a s ⊆ a ′ , and define a ring A to be S –noetherian whenever every ideal a ⊆ A is S –finite.Many authors have worked on S –noetherian rings and related notions, and shown rele-vant results about its structure. See for instance [
2, 5, 9, 10, 11, 13 ] .The main aim of this paper is to give a new approach to S –noetherian rings and modules,and its applications, using the more abstract notion of hereditary torsion theory. From [email protected]; November 6, 2020 A , see [
12, 3 ] , and we denote by Mod − A the category of A –modules. Thus,a hereditary torsion theory σ in Mod − A is given by one of the following objects:(1) a torsion class T σ , a class of modules which is closed under submodules, homomor-phic images, direct sums and group extensions,(2) a torsionfree class F σ , a class of modules which is closed under submodules, es-sential extensions, direct products and group extensions,(3) a Gabriel filter of ideals L ( σ ) , a non–empty filter of ideals satisfying that every b ⊆ A , for which there exists an ideal a ∈ L ( σ ) such that ( b : a ) ∈ L ( σ ) , for every a ∈ a , belongs to L ( σ ) .(4) a left exact kernel functor σ : Mod − A −→ Mod − A .The relationships between these notions are the following. If σ is the left exact kernelfunctor, then T σ = { M ∈ Mod − A | σ M = M } , F σ = { M ∈ Mod − A | σ M = } , L ( σ ) = { a ⊆ A | A / a ∈ T σ } .If L is the Gabriel filter of a hereditary torsion theory σ , and T is the torsion class, forany A –module M we have: σ M = { m ∈ M | ( m ) ∈ L } = X { N ⊆ M | N ∈ T } . Example 0.1. (1) Let Σ ⊆ A be a multiplicatively closed subset, there exists a hereditarytorsion theory, σ Σ , defined by L ( σ Σ ) = { a ⊆ A | a ∩ Σ = ∅ } .Observe that σ Σ has a filter basis constituted by principal ideals. A hereditary torsiontheory σ such that L ( σ ) has a filter basis of principal ideals is called a principalhereditary torsion theory . We can show there is a correspondence between prin-cipal hereditary torsion theories in Mod − A , and saturated multiplicatively closedsubsets in A .(2) For any ring A the set L = { a ⊆ A | Ann ( a ) = } is a Gabriel filter, it defines ahereditary torsion theory which we represent by λ .This paper is organized in sections. In the first one we introduce totally σ –noetherianrings and modules and show that necessarily the hereditary torsion theory σ is of finitetype whenever the ring A is totally σ –noetherian. In section two we study how prime2deals and prime submodules appears naturally when studying totally σ –finitely gener-ated modules and obtain a relative version of Cohen’s theorem. The natural notion ofmaximal condition in relation with totally σ –noetherian modules is studied in sectionthree. Section four is devoted to study some extensions of totally σ –noetherian rings.In particular, we find that it is necessary to impose some extra conditions to σ in orderto assure that A [ X ] is totally σ –noetherian whenever A is. The local behaviour of totally σ –noetherian modules is studied in section four in which we can reduce to consider onlyprime ideals in K ( σ ) . In the last section we study the particular case of principal idealrings.Through this paper we try to study σ –noetherian rings and modules and, in a parallel way,totally σ -noetherian rings and modules. The first one ( σ –noetherian) has a categoricalbehaviour, but not the second one (totally σ –noetherian). For that reason, the studyof the last one is more difficult and it is not in the literature. Otherwise, it producesresults as Proposition (5.2.) which shows that to be totally σ –noetherian, as opposed to σ -noetherian, is a local property. σ –noetherian rings and modules For any σ –torsion finitely generated A –module M , say M = m A + · · · + m t A , since ( m i ) ∈ L ( σ ) , for any i =
1, . . . , t , if h : = ∩ ti = ( m i ) ∈ L ( σ ) , they satisfy M h = σ –torsion non–finitely generated A –modules.Therefore, we shall define an A –module M to be totally σ –torsion whenever there exists h ∈ L ( σ ) such that M h =
0. This notion of totally torsion appears, for instance, in [ ] .For any ideal a ⊆ A we have two different notions of finitely generated ideals relative to σ :(1) a ⊆ A is σ –finitely generated whenever there exists a finitely generated ideal a ′ ⊆ a such that a / a ′ is σ –torsion.(2) a ⊆ A is totally σ –finitely generated whenever there exists a finitely generated ideal a ′ ⊆ a such that a / a ′ is totally σ –torsion.In the same way, for any ring A we have two different notions of noetherian ring relativeto σ :(1) A is σ –noetherian if every ideal is σ –finitely generated.(2) A is totally σ –noetherian whenever every ideal is totally σ –finitely generated. Example 1.1. (1) Every finitely generated ideal is totally σ –finitely generated and everytotally σ –finitely generated ideal is σ –finitely generated.32) Let S ⊆ A be a multiplicatively closed subset, an ideal a ⊆ A is S –finite if, and onlyif, it is totally σ S –finitely generated. The ring A is S –noetherian if, and only if, A istotally σ S –noetherianThese two notions of torsion, and the notions derived from them, are completely differentin its behaviour and its categorical properties. For instance, due to the definition, forany A –module M there exists a maximum submodule belonging to T σ , the submodule: σ M , and it satisfies M /σ M ∈ F ( σ ) . In the totally σ –torsion case we can not assurethe existence of a maximal totally σ –torsion submodule. The existence of a maximum σ –torsion submodule allows us to build new concepts relative to σ as lattices, closureoperators and localizations; concepts that we have not in the totally σ –torsion case.Nevertheless, the totally σ –torsion case allows us to study arithmetic properties of ringsand modules which are hidden with that use of σ –torsion, and these properties are thosewhich we are interested in studying.As we pointed out before, the σ –torsion allows, for any A –module M , to define a lattice C ( M , σ ) = { N ⊆ M | M / N ∈ F σ } ,and a closure operator Cl M σ ( − ) : L ( M ) −→ C ( M , σ ) ⊆ L ( M ) , from L ( M ) , the lattice ofall submodules of M , defined by the equation Cl M σ ( N ) / N = σ ( M / N ) . The elements in C ( M , σ ) are called the σ –closed submodules of M , and the lattice operations in C ( M , σ ) ,for any N , N ∈ C ( M , σ ) , are defined by N ∧ N = N ∩ N , N ∨ N = Cl M σ ( N + N ) .Dually, the submodules N ⊆ M such that M / N ∈ T σ are called σ –dense submodules.The set of all σ –dense submodules of M is represented by L ( M , σ ) .In the following, we assume A is a ring, Mod − A is the category of A –modules and σ is a hereditary torsion theory on Mod − A . Modules are represented by Latin letters: M , N , N , . . ., and ideals by Gothics letters: a , b , b , . . . Different hereditary torsion theorieswill be represented by Greek letters: σ , τ , σ , . . ., and induced hereditary torsion theoriesby adorned Greek letters: σ ′ , τ , . . .The notions of (totally) σ –finitely generated and (totally) σ –noetherian can be extendedto A –modules in an easy way. Properties on the behaviour of totally σ –finitely generatedand σ –noetherian modules are collected in the following result. Proposition 1.2. (1) Every homomorphic image of a totally σ –finitely generated A –modulealso is.(2) For every submodule N ⊆ M , we have: M is totally σ –noetherian if, and only if, N and M / N are totally σ –noetherian. 43) Finite directsumsoftotally σ –noetherianmodulesalso are.The first restriction we have found in studying totally σ –noetherian rings is that thehereditary torsion theory σ must be of an special type: it is a finite type hereditarytorsion theory. This is an extension of principal hereditary torsion theories, and meansthat L ( σ ) has a filter basis constituted by finitely generated ideals. Proposition 1.3. If A isatotally σ –noetherianring then σ isoffinite type.P ROOF
For any a ∈ L ( σ ) there exists a ′ ⊆ a , finitely generated, and h ∈ L ( σ ) such that ah ⊆ a ′ . Since L ( σ ) is closed under product of ideals, we have h ′ ∈ L ( σ ) . ƒ Directly from the definition we have that every totally σ –torsion module is totally σ –noetherian, and an A –module M is totally σ –noetherian if, and only if it contains a finitelygenerated submodule N ⊆ M such that M / N is totally σ –torsion. Our aim is to exploremore conditions equivalent to totally σ –noetherian. In this sense, since every totally σ –noetherian module is σ –noetherian, the question is what properties are necessary to addto σ –noetherianness to get totally σ –noetherian.The next proposition is based on [
1, Proposition 2 ] . Proposition 1.4.
Let σ beafinitetypehereditarytorsiontheory,and M bean A –module,the followingstatements areequivalent:(a) M istotally σ –noetherian.(b) M is σ –noetherian, and forevery H ⊆ M , finitelygenerated, thereexists h ∈ L ( σ ) ,finitelygenerated,such that Cl M σ ( H ) = ( H : h ) .P ROOF (a) ⇒ (b). If Cl M σ ( N ) is totally σ –finitely generated, there exist H ⊆ Cl M σ ( N ) ,finitely generated, and h ∈ L ( σ ) , finitely generated, such that Cl M σ ( N ) h ⊆ H ⊆ Cl M σ ( N ) .There exists b ∈ L ( σ ) , finitely generated, such that H b ⊆ N . Therefore, N hb ⊆ Cl M σ ( N ) hb ⊆ H hb ⊆ H b ⊆ N .In particular, Cl M σ ( N ) ⊆ ( N : hb ) . On the other hand, ( N : hb ) hb ⊆ N , hence Cl M σ ( N ) =( N : hb ) .(b) ⇒ (a). Let N ⊆ M , there is H ⊆ N , finitely generated, such that Cl M σ ( N ) = Cl M σ ( H ) ,and there exists h ∈ L ( σ ) , finitely generated such that Cl M σ ( H ) = ( H : h ) . Therefore wehave: N h ⊆ Cl M σ ( N ) h = Cl M σ ( H ) h ⊆ H ⊆ N . ƒ We know how to induce hereditary torsion theories through a ring map. Here we shallstudy the particular case of a ring map f : A −→ B such that every ideal of B is extendedof an ideal of A , i.e., for any ideal b ⊆ B there exists an ideal a ⊆ A such that f ( a ) B = b σ be a hereditary torsion theory in Mod − A , then f ( σ ) is a hereditary torsion theoryin Mod − B and its Gabriel filter is L ( f ( σ )) = { b ⊆ B | f − ( b ) ∈ L ( σ ) } .It is clear that f ( σ ) is of finite type whenever σ is.In this situation we have: Lemma 1.5.
Let σ be a finite type hereditary torsion theory, f : A −→ B be a ring mapsuch that every ideal of B is an extended ideal. In this case L ( f ( σ )) = { f ( a ) B | a ∈L ( σ ) } .If A istotally σ –noetherian, then B istotally f ( σ ) –noetherian.P ROOF
Let b ⊆ B , there exists a ⊆ A such that b = f ( a ) B . There exists h ∈ L ( σ ) , finitelygenerated, such that ah ⊆ a ′ ⊆ a , for some finitely generated ideal a ′ ⊆ a . Therefore, b f ( h ) B = f ( a ) f ( h ) B ⊆ f ( a ′ ) B ⊆ b . ƒ Examples of this situation are the following:(1) B is the quotient of a ring A by an ideal a , i.e., p : A −→ A / a .(2) B is the localized ring of A at a multiplicatively closed subset Σ ⊆ A , i.e., q : A −→ Σ − A . If σ is a hereditary torsion theory in Mod − A , it is well known that for any prime ideal p ⊆ A we have either p ∈ C ( A , σ ) or p ∈ L ( σ ) , i.e., either A / p is σ –torsionfree or A / p is σ –torsion. In consequence, σ produces a partition of Spec ( A ) in two sets: Spec ( A ) = K ( σ ) ∪Z ( σ ) , with K ( σ ) ⊆ C ( A , σ ) , and Z ( σ ) ⊆ L ( σ ) . In addition, for every p ∈ K ( σ ) we have σ ≤ σ A \ p , and σ = ∧{ σ A \ p | p ∈ K ( σ ) } , whenever σ is of finite type. Proposition 2.1.
Let σ be a finite type hereditarytorsiontheoryin Mod − A , and let M be a totally σ –finitely generated module. If N ⊆ M is maximal among all non–totally σ –finitely generated submodulesof M , then ( N : M ) isaprimeideal.P ROOF
Let p = ( N : M ) ; if p is not prime, there exist a , b ∈ A \ p such that a b ∈ p .As a consequence, since N $ N + M b , then N + M b is totally σ –finitely generated, and N , M a ⊆ ( N : b ) , hence N $ N + M a ⊆ ( N : b ) , then ( N : b ) is totally σ –finitelygenerated.Let h = 〈 h , . . . , h s 〉 ∈ L ( σ ) such that ( N + M b ) h ⊆ F = 〈 f , . . . , f t 〉 ⊆ N + M b , and ( N : b ) h ⊆ G = 〈 g , . . . , g r 〉 ⊆ ( N : b ) . Let f j = n j + m j b for any j =
1, . . . , t , where n j ∈ N and m j ∈ M . 6or any n ∈ N and any h i ∈ h there exists an A –linear combination nh i = P j ( n j + m j b ) c i , j ,with c i , j ∈ A . Let x i = P j m j c i , j ; therefore x i b = P j m j c i , j b = na i + P j n j c i , j ∈ N , hence x i ∈ ( N : b ) .Let us represent now the generators of h by h ′ l . For any h ′ l we have x i h ′ l ∈ G , and thereexists an A –linear combination x i h ′ l = P k g k d l , i , k , with d l , i , k ∈ A . Thus we have: nh i h ′ l = ( nh i ) h ′ l = ( P j n j c i , j + P j m j c i , j b ) h ′ l = P j n j c i , j h ′ l + P j m j c i , j bh ′ l = P j ( n j h ′ l ) c i , j + x i bh ′ l = P j ( n j h ′ l ) c i , j + ( P k g k d l , i , k ) b = P j ( n j h ′ l ) c i , j + P k ( g k b ) d l , i , k ∈ 〈 n j h ′ l , g k b | l , j =
1, . . . , s ; k =
1, . . . , r 〉 ⊆ N .In particular, N hh ⊆ 〈 n j h ′ l , g k b | l , j =
1, . . . , s ; k =
1, . . . , r 〉 ⊆ N . This means that N istotally σ –finitely generated, which is a contradiction. ƒ If N ⊆ M is maximal among the non totally σ –finitely generated submodules, is it aprime submodule? We know that it holds in the case of finitely generated modules. Letus prove it now for totally σ –finitely generated modules. Proposition 2.2.
Let σ be afinite type hereditarytorsiontheory,and M be atotally σ –finitely generated A –module. Any N ⊆ M , maximal among the submodules of M whichare not totally σ –finitelygenerated,isaprimesubmodule.P ROOF
Let N ⊆ M such a maximal submodule. If N ⊆ M is not prime, there exist m ∈ M \ N and a ∈ A \ ( N : M ) such that ma ∈ N .Since a / ∈ ( N : M ) , then M a * N , and N $ N + M a is totally σ –finitely generated. Onthe other hand, since m ∈ ( N : a ) \ N , then N $ ( N : a ) is totally σ –finitely generated.Therefore, there exist a finitely generated submodules F = 〈 f , . . . , f r 〉 ⊆ N + M a and G = 〈 g , . . . , g s 〉 ⊆ ( N : a ) , and h = 〈 h , . . . , h t 〉 ∈ L ( σ ) , finitely generated, such that ( N + M a ) h ⊆ F ⊆ N + M a and ( N : a ) h ⊆ G ⊆ ( N : a ) . Say f i = n i + m i a for i =
1, . . . , r , n i ∈ N and m i ∈ M .For any n ∈ N and h i ∈ { h , . . . , h t } , since nh i ∈ F , there exists an A –linear combination nh i = P l f l c i , l = P l n l c i , l + P l m h c i , l a , and ( P l m l c i , l ) a = n − P l n l c i , l ∈ N . In conse-quence P l m l c i , l ∈ ( N : a ) .For any h j ∈ { h , . . . , h t } we have ( P l m l c i , l ) h j ∈ G , and there exists an A –linear combi-nation ‚X l m l c i , l Œ h j = X k g k d i , j , k , with d i , j , k ∈ A .7herefore, we have nh i h j = ‚X l n l c i , l + X l m l c i , l a Œ h j = X l n l c i , l h j + X l m l c i , l h j a = X l n l c i , l h j + X k g k d i , j , k a ,which means that n hh is contained in the submodule of N generated by { n , . . . , n r } and { g a , . . . , g s a } . ƒ The next one is a properly result of finite type hereditary torsion theories.
Lemma 2.3.
Let σ beafinitetypehereditarytorsiontheory. Foreverytotally σ –finitelygenerated A –module M , and any L ∈ C ( M , σ ) , L $ M , there exists a maximal element N ∈ C ( M , σ ) such that L ⊆ N . In addition, if Γ = { N ⊆ M | L ⊆ N ∈ C ( M , σ ) , N = M } ,everymaximal elementin Γ isaprimesubmodule of M .P ROOF
Let { N i } i be a chain in Γ , and we define N = ∪ i ∈ I N i . If Cl M σ ( N ) = M , then Cl M σ ( N ) is an upper bound of the chain in Γ . If Cl M σ ( N ) = M , since there exist m , . . . , m t ∈ M and h ∈ L ( σ ) , finitely generated, such that M h ⊆ ( m , . . . , m t ) A ⊆ M , then there exists b ∈ L ( σ ) , finitely generated, such that ( m , . . . , m t ) b ⊆ ∪ i N i = N ; therefore, there existsan index i such that ( m , . . . , m t ) b ⊆ N i . In consequence, N i = Cl M σ ( N i ) = M , which is acontradiction. ƒ The following result is based in [
8, Theorem 1 ] , see also [
1, Proposition 4 ] . Proposition 2.4.
Let σ be a finite type hereditarytorsiontheoryin Mod − A , and let M be atotally σ –finitelygenerated module. Thefollowingstatementsare equivalent:(a) M istotally σ –noetherian.(b) For every prime ideal p ∈ K ( σ ) the submodule M p ⊆ M is totally σ –finitely gener-ated.P ROOF
Clearly, if p ∈ Z ( σ ) , then M p ⊆ M is σ –dense, hence totally σ –finitely gener-ated, because M is.(a) ⇒ (b). It is obvious.(b) ⇒ (a). Since M is totally σ –finitely generated, there exists h ∈ L ( σ ) , finitely gener-ated, such that h M ⊆ ( a , . . . , a t ) ⊆ M . If M is not totally σ –noetherian, Γ = { N ⊆ M | N is not totally σ –finitely generated } is not empty. Any chain in Γ has a upper bound in Γ , hence, by Zorn’s lemma, there exists N ∈ Γ maximal. The ideal p = ( N : M ) is prime; if p ∈ Z ( σ ) , then N is totally σ –finitely8enerated as M is. Therefore, p ∈ K ( σ ) , and, by the hypothesis, M p is totally σ –finitelygenerated, there exists h ′ ∈ L ( σ ) such that M ph ′ ⊆ ( b , . . . , b s ) ⊆ M p .Since p = ( N : M ) ⊆ ( N : ( a , . . . , a t )) ⊆ ( N : M h ) = ( p : h ) = p , as p is prime, the p = ( N : a ) ∩ . . . ∩ ( N : a t ) , hence there exists an index i such that p = ( N : a i ) , and a i / ∈ N . Therefore N + Aa i % N , and N + Aa i is totally σ –finitely generated. There exists h ′′ ∈ L ( σ ) such that N h ′′ ⊆ ( n + x a i , . . . , n r + x r a i ) ⊆ N + Aa i , where x , . . . , x r ∈ A ,and N h ′′ ⊆ ( n , . . . , n r ) + a i p . Then N h ′′ h ′ ⊆ ( n , . . . , n r ) h ′ + ( b , . . . , b s ) ⊆ N + M p ⊆ N ,which is a contradiction. ƒ As a direct consequence we have:
Corollary 2.5. (Cohen’s theorem)
Let σ be a finite type hereditary torsion theory in Mod − A ,the followingstatementsare equivalent:(a) A istotally σ –noetherian.(b) Everyprimeideal in K ( σ ) istotally σ –finitelygenerated.P ROOF
It is a direct consequence of Proposition (2.4.). ƒ When we particularize to the hereditary torsion theory σ =
0, i.e, when L ( σ ) = { A } ,we have that A is a noetherian ring if, and only if, every prime ideal is finitely generated,which is Cohen’s Theorem. Otherwise, if σ = σ S , for some multiplicatively closed subset S ⊆ A , then A is S –noetherian if, and only if, every prime ideal, in K ( σ S ) , is totally S –finite, see [ ] . Let M be an A –module, an increasing chain of submodules N ⊆ N ⊆ · · · is totally σ –stable whenever there exist an index m and h ∈ L ( σ ) such that N m h ⊆ N s for every s ≥ m . Proposition 3.1.
Let σ be a hereditary torsion theory and M be an A –module. Thefollowingstatementsare equivalent:(a) M istotally σ –noetherian.(b) Everyincreasing chain N ⊆ N ⊆ · · · istotally σ –stable.P ROOF (a) ⇒ (b). Let N ⊆ N ⊆ · · · be an increasing chain of submodules of M , anddefine N = ∪ s ≥ N s . By the hypothesis, there exist h ⊆ L ( σ ) and n , . . . , n t ∈ N such that N h ⊆ ( n , . . . , n t ) ⊆ N . Therefore, there exists an index m such that n , . . . , n t ∈ N m , andwe have N h ⊆ N m . In particular, N s h ⊆ N m for every s ≥ m .(b) ⇒ (a). Let N ⊆ M , for any increasing chain { N s } s ∈ N of totally σ –finitely generatedsubmodules of N there exist an index m and h ∈ L ( σ ) such that N s h ⊆ N m , for any s ≥ m .9n particular, N h ⊆ N m , and N is totally σ –finitely generated. This means that the familyof all totally σ –finitely generated submodules of N has maximal elements. Let H ⊆ N one of these maximal elements. If H $ N , there exists n ∈ N \ H , hence H + nA ⊆ N istotally finitely generated, which is a contradiction. ƒ Let M be an A –module, we have the following definitions:(1) Let N ⊆ L ( M ) be a family of submodules of M . An element N ∈ N is σ –maximal ifthere exists h ∈ L ( σ ) such that for every H ∈ N satisfying N ⊆ H we have H h ⊆ N .(2) The A –module M satisfies the σ -MAX condition if every nonempty family of sub-modules of M has σ –maximal elements.(3) A family N of submodules of M is σ –upper closed if for every submodule H ⊆ M such that there exist N ∈ N and h ∈ L ( σ ) satisfying H and H h ⊆ N , or equivalently H ⊆ ( N : h ) , we have H ∈ N . Proposition 3.2.
Let M bean A –module,thefollowingstatements are equivalent:(a) M istotally σ –noetherian.(b) Everynonempty σ –upperclosedfamilyofsubmodulesof M hasmaximalelements.(c) Everynonemptyfamilyofsubmodulesof M has σ –maximal elements.P ROOF (a) ⇒ (b). Let N be a nonempty σ –upper closed family of submodules of M . Forany increasing chain N ⊆ N ⊆ · · · in N we define N = ∪ s ≥ N s . By the hypothesis thereexist an index m and h ∈ L ( σ ) such that N s h ⊆ N m , for every s ≥ m . Hence N h ⊆ N m ,and N ∈ N . In consequence, by Zorn’s lemma, N contains maximal elements.(b) ⇒ (c). Let N be a nonempty family of submodules of M . We define a new family N = { H ⊆ M | there exist N ∈ N and h ∈ L ( σ ) such that H h ⊆ N } ,the σ –upper closure of N . We claim N is σ –upper closed. Indeed, if L ⊆ M , H ∈ N and h ∈ L ( σ ) satisfy L ⊆ ( H : h ) , by the hypothesis there exist N ∈ N and h ′ ∈ L ( σ ) such that H ⊆ ( N : h ′ ) , hence we have L ⊆ ( H : h ) ⊆ (( N : h ′ ) : h ) = ( N : h ′ h ) , and L ∈ N .By the hypothesis, there exists a maximal element, say H , in N , and there exist N ∈ N and h ∈ L ( σ ) such that N ⊆ H ⊆ ( N : h ) . Since ( N : h ) ∈ N , then H = ( N : h ) . We claim N is σ –maximal in N . Indeed, if N ⊆ L for some L ∈ N , then H = ( N : h ) ⊆ ( L : h ) , bythe maximality of H we have ( L : h ) = ( N : h ) , hence L h ⊆ N .(c) ⇒ (a). Let N ⊆ N ⊆ · · · be an increasing chain of submodules of M . We considerthe family N = { N s | s ∈ N \ { }} . By the hypothesis N has σ –maximal elements. If N m ∈ N is σ –maximal, there exists h ∈ L ( σ ) such that N s h ⊆ N m for every s ≥ m ; hence M is totally σ –noetherian. ƒ Observe that if M is an A –module, for any submodule N ⊆ M we may consider the family N = { N } , and its σ –upper closure N = { H ⊆ M | there exists h ∈ L ( σ ) such that N ⊆ H ⊆ ( N : h ) } ,hence for every H ∈ N we have N ⊆ H ⊆ Cl M σ ( N ) . In addition, we have:10 emma 3.3. N has onlyonemaximal elementif, and onlyif, Cl M σ ( N ) ∈ N .P ROOF
Let H ∈ N be the only maximal element, if there exists x ∈ Cl M σ ( N ) \ H , thereare h ∈ L ( σ ) such that H h ⊆ N and x h ⊆ N , hence ( H + ( x )) h ⊆ N , and H + ( x ) ∈ N ,which is a contradiction. ƒ Lemma 3.4.
Let M be an A –module and T ⊆ M be a totally σ –torsion submodule, thefollowingstatementsare equivalent:(a) M istotally σ –noetherian.(b) M / T istotally σ –noetherian.P ROOF (a) ⇒ (b). Let N / T ⊆ N / T ⊆ · · · be an increasing chain of submodules of M / T , there exist m ∈ N and h ∈ L ( σ ) such that N s h ⊆ N m , for every s ≥ m . Therefore N s T h = N s h + TT ⊆ N m T .(b) ⇒ (a). Let N ⊆ N ⊆ · · · be an increasing chain of submodules of M , then ( N + T ) / T ⊆ ( N + T ) / T ⊆ · · · is an increasing chain of submodules of M / T , and there exist m ∈ N , h ∈ L ( σ ) such that N s + TT h ⊆ N m + TT , for every s ≥ m . Otherwise, there exists h ′ ∈ L ( σ ) such that T h ′ =
0. Therefore, N s hh ′ = ( N s h + T ) h ′ ⊆ ( N m + T ) h ′ = N m h ′ ⊆ N m . ƒ Let σ be a hereditary torsion theory in Mod − A , and f : A −→ B be a ring map. Lemma 4.1.
The set L ( f ( σ )) = { b ⊆ B | f − ( b ) ∈ L ( σ ) } isa Gabriel filterin B , and itdefines ahereditarytorsiontheoryin Mod − B , being(1) T f ( σ ) = { M B | M A ∈ T σ } and(2) F f ( σ ) = { M B | M A ∈ F σ } .We name f ( σ ) the hereditary torsion theory induced by σ through the ring map f . Lemma 4.2.
Let σ beafinitetypehereditarytorsiontheoryin Mod − A ,and f : A −→ B be aringmap,the induced hereditarytorsiontheory f ( σ ) isoffinite type.P ROOF
Let b ∈ L ( f ( σ )) , there exists a ∈ L ( σ ) , finitely generated, such that a ⊆ f − ( b ) .Therefore, f ( a ) B ∈ L ( f ( σ )) is finitely generated, and f ( a ) ⊆ b , hence f ( σ ) is finitelygenerated. ƒ orollary 4.3. (Eakin–Nagata theorem) Let σ beafinitetypehereditarytorsiontheoryin Mod − A ,and f : A , → B bearingextensionsuchthat B istotally σ –finitelygeneratedand B p istotally f ( σ ) –finitelygeneratedforeveryprimeideal p ∈ K ( σ ) ,then A istotally σ –noetherian.P ROOF
It is sufficient to prove that B is a totally σ –noetherian A –module because A issubmodule of B , or equivalently that for every prime ideal p ∈ K ( σ ) we have that p B ⊆ B is totally σ –finitely generated. There exists a ∈ L ( σ ) such that ap B ⊆ ( p , . . . , p t ) B ⊆ p B ,and there exists a ′ ∈ L ( σ ) such that a ′ B ⊆ ( b , . . . , b s ) A ⊆ A , hence a ′ ap B ⊆ a ′ ( p , . . . , p t ) B ⊆ ( p , . . . , p t )( b , . . . , b s ) A . ƒ Corollary 4.4. (Eakin–Nagata theorem)
Let σ beafinitetypehereditarytorsiontheoryin Mod − A , f : A , → B bearingextensionsuchthat B istotally σ –finitelygeneratedandtotally f ( σ ) –noetherian,then A istotally σ –noetherian. Proposition 4.5.
Let σ be a finite type hereditary torsion theory in Mod − A , and f : A , → B bearingextensionsuchthat B isfaithfullyflat( a B ∩ A = a foreveryideal a ⊆ A ).If B istotally f ( σ ) –noetherian,then A istotally σ –noetherian.P ROOF
Let a ⊆ A , since a B ⊆ B is totally f ( σ ) –finitely generated, there exists c ∈ L ( σ ) such that ca B ⊆ ( b , . . . , b t ) B , for some b , . . . , b t ∈ a . Therefore, ca = c ( a B ∩ A ) ⊆ ( b , . . . , b t ) B ∩ A = ( b , . . . , b t ) A . ƒ In order to consider polynomial extensions, we introduce a new kind of finite type hered-itary torsion theories. A finite type hereditary torsion theory σ in Mod − A is almostjansian or anti–archimedian if for every ideal a ∈ L ( σ ) we have ∩ ∞ n = a n ∈ L ( σ ) . Examples 4.6. (1) An example of almost jansian hereditary torsion theories are the jan-sian. A hereditary torsion theory σ is jansian whenever L ( σ ) has a filter basis con-stituted by an ideal a ; in this case a must be idempotent. If in addition, σ is of finitetype then a is finitely generated, hence generated by an idempotent element, say e ∈ A , and the localization of A at σ is just the ring eA .(2) A multiplicatively closed subset Σ ⊆ A is anti–archimedean whenever ∩ ∞ n = a n A ∩ Σ = ∅ , hence if, and only if, σ Σ is almost jansian.(3) An integral domain D is anti–archimedean if ∩ ∞ n = a n D = a ∈ D , hence wecan rewrite D is anti–archimedean if, and only if, σ D \{ } is almost jansian.(4) Let p ⊆ A be a prime ideal, the hereditary torsion theory σ A \ p is of finite type, andit is almost jansian if, and only if, for any ideal a ⊆ A if a * p , then ∩ n a n * p . Inparticular, if, and only if, A / p is an anti–archimedean domain; let us call such a p an anti–archimedean prime ideal of A . 125) For every strongly prime ideal p ⊆ A , see [ ] , the hereditary torsion theory σ A \ p isalmost jansian.(6) Since the intersection of finitely many finite type hereditary torsion theories is offinite type, if { p , . . . , p t } are anti–archimedean prime ideals of A , then ∧ ti = σ A \ p i isalmost jansian. Theorem 4.7. (Hilbert basis theorem)
Let σ beafinitetypealmostjansian hereditarytorsiontheoryin Mod − A , and σ ′ theinduced hereditarytorsiontheoryin Mod − A [ X ] .If A istotally σ –noetherian, then A [ X ] istotally σ ′ –noetherian.P ROOF
Let b ⊆ A [ X ] be an ideal, we define a = { lc ( F ) | F ∈ b } , being, for convenience,lc ( ) =
0. Thus a ⊆ A is an ideal, and there exist h ∈ L ( σ ) , finitely generated, and a , . . . , a t ∈ a such that ah ⊆ ( a , . . . , a t ) A ⊆ a . Let F , . . . , F t ∈ b such that lc ( F i ) = a i forany i ∈ {
1, . . . , t } , and d = max { deg ( F i ) | i ∈ {
1, . . . , t }} .For any n ∈ N \ { } , we define H n = { F ∈ b | deg ( F ) < n } , thus H n is an A –moduleisomorphic to a submodule of the free A –module A n , hence it is totally σ –finitely gener-ated. There exist h n ∈ L ( σ ) , finitely generated, and H , . . . , H s ∈ H n such that H n h n ⊆ ( H , . . . , H s ) A ⊆ H n .When we take H d , we may assume that h n and h are equal. Let us suppose that h =( h , . . . , h r ) A .Let F ∈ b . If f = deg ( F ) < d , then F ∈ H d . If f = deg ( F ) ≥ d , we have lc ( F ) ∈ a , and lc ( F ) h ⊆ ( a , . . . , a t ) A . For any j ∈ {
1, . . . , r } there exists an A –linear combi-nation lc ( F ) h j = P ti = a i c i , j . Hence there exist natural numbers e , . . . , e t such that F h j − P ti = F i X e i c i , j = G j is a polynomial in b of degree less than f , i.e., G j ∈ H f . Since F h j = P ti = F i c i , j + G j , then F h ⊆ ( F , . . . , F t ) A + ( G , . . . , G r ) A . We resume this fact sayingthat for any 0 = F ∈ b of degree f ≥ d , there exists a finite subset G ⊆ H f such that F h ⊆ ( F , . . . , F t ) A + G A ; if f < d , then G = { H , . . . , H s } ⊆ H d .Starting from a polynomial F ∈ b of degree f ≥ d there exists a finite subset G ⊆ H f such that f h ⊆ ( F , . . . , F t ) A + G A . For any G ∈ G there exists G ′ ⊆ H f − such that G h ⊆ ( F , . . . , F t ) A + G ′ A , hence there exists a finite subset G ⊆ H f − such that F h ⊆ ( F , . . . , F t ) A + G A . And, iterating this process, there exists k ∈ N and a finite subset G ⊆ H d such that F h k ⊆ ( F , . . . , F t ) A + G A .In consequence, b ( ∩ k ∈ N h k ) ⊆ ( F , . . . , F t ) + ( G ) ⊆ b , and since σ is almost jansian ∩ k h k ∈L ( σ ) , then b is totally σ ′ –finitely generated. ƒ The following consequence also holds.
Corollary 4.8.
Let σ beafinitetypealmostjansianhereditarytorsiontheoryin Mod − A ,and σ ′ the induced hereditary torsion theory in Mod − A [ X , . . . , X n ] . If A is totally σ –noetherian, then A [ X , . . . , X n ] istotally σ ′ –noetherian.13lso we can prove that totally σ –noetherianness is preserved by localization at multi-plicatively subsets. Proposition 4.9.
Let A bearingand σ beafinitetypehereditarytorsiontheoryin Mod − A . If Σ ⊆ A isamultiplicativelysubset,andwedenoteby σ ′ theinducedhereditarytorsiontheoryin Mod − A Σ ,then A Σ istotally σ ′ –noetherian.P ROOF
Let g : A −→ A Σ be the canonical ring map. Let p ⊆ A Σ be a prime ideal, then g − ( b ) ⊆ A is prime. By the hypothesis there exist h ∈ L ( σ ) and b , . . . , b t ∈ g − ( b ) suchthat g − ( b ) h ⊆ ( b , . . . , b t ) A ⊆ g − ( b ) . If we localize at Σ , then ( g − ( b ) h ) Σ ⊆ ( b , . . . , b t ) A Σ ⊆ g − ( b ) Σ ,i.e., bh Σ ⊆ ( b , . . . , b t ) A Σ ⊆ b .Therefore, b is totally σ ′ –finitely generated. ƒ As a consequence we have:
Corollary 4.10.
Let A bearingand σ beanalmostjansianfinitetypehereditarytorsiontheorysuch that A istotally σ –noetherian, if σ ′ is the induced hereditarytorsiontheoryin A [ X , X − ] , then A [ X , X − ] istotally σ ′ –noetherian.P ROOF
Let A f −→ A [ X ] g −→ A [ X , X − ] . If we consider Σ = { X t | t ∈ N } ⊆ A [ X ] ,then A [ X , X − ] = A [ X ] Σ . Since σ is almost jansian then A is totally σ –noetherian, then A [ X ] is totally f ( σ ) –noetherian, and A [ X , X − ] is totally g f ( σ ) –noetherian, by Proposi-tion (4.9.). ƒ Theorem 4.11. (Hilbert basis theorem)
Let σ beafinitetypealmostjansianhereditarytorsiontheoryin Mod − A ,and σ ′ theinducedhereditarytorsiontheoryin Mod − A ¹ X º .If A istotally σ –noetherian, then A ¹ X º istotally σ ′ –noetherian.P ROOF
Let a ⊆ A ¹ X º be an ideal. For any n ∈ N we define a n = 〈 first coefficient of a F ∈ a ∩ ( X n ) 〉 . Hence we obtain an increasing chain a ⊆ a ⊆ · · · . Since A is totally σ –noetherian, there exist m ∈ N and h ∗ ∈ L ( σ ) , finitely generated, such that a s h ∗ ⊆ a m for any s ≥ m . Otherwise, for any a n there exists a finitely generated ideal a ′ n ⊆ a n , and h n ∈ L ( σ ) such that a n h n ⊆ a ′ n . If we take h = h ∗ h · · · h m , then a n h ⊆ a ′ n , for any n ≤ m a n h ⊆ ah ∗ h m ⊆ a m h m ⊆ a ′ m , for any n > m .For any n =
0, 1, . . . , m , if a ′ n = 〈 a n ,1 , . . . , a n , t n 〉 , we take F i , j ∈ a ∩ ( X n ) such that the firstcoefficient of F i , j is a i , j . 14or any F ∈ a we have: F h ∈ ( a ′ + a ′ X + · · · ) ∩ a ∩ ( X n ) . If F = ( b + b X + · · · ) X n ∈ a ∩ ( X n ) ,then F h ∈ ( a ′ n X n + · · · ) ∩ a ∩ ( X n ) , and, for every h ∈ h there exists a linear combination b h = P t n j = a n , j c n , j , hence F h − P t n j = F n , j c n , j ∈ a ∩ ( X n + ) . In the same way, for every h ′ ∈ h there exists a linear combination F hh ′ − P t n j = F n , j c n , j h ′ − P t n + j = F n + j c n + j ∈ a ∩ ( X n + ) .This reduces the problem to study what happens with the elements in a ∩ ( X m ) . As wesaw before, if F ∈ a ∩ ( X m ) , for any h ∈ h there exists a linear combination F h − P t m j = F m , j c m , j ∈ a ∩ ( X m + ) . Since a m + h ⊆ a ′ m , for any h ∈ h there exists a linearcombination F h h − P t m j = F m , j c m , j h − P t m j = F m , j c m , j ∈ a ∩ ( X m + ) ; therefore, F h h − P t m j = F m , j ( c m , j h + c m , j ) ∈ a ∩ ( X m + ) . In general, F h h · · · h s − t m X j = F m , j ( c m , j h · · · h s + c m , j h · · · h s + · · · + c s , m , j ) ∈ a ∩ ( X m + s ) .Since σ is almost jansian, i.e., ∩ n h n ∈ L ( σ ) , if h = ∩ n h n , then F h ⊆ 〈 F m ,1 , . . . , F m , t m 〉 . ƒ Corollary 4.12.
Let σ beafinitetypealmostjansianhereditarytorsiontheoryin Mod − A ,and σ ′ the induced hereditary torsion theory in Mod − A ¹ X , . . . , X n º . If A is totally σ –noetherian, then A ¹ X , . . . , X n º istotally σ ′ –noetherian. Let p ⊆ A be a prime ideal, we consider σ A \ p , the hereditary torsion theory cogeneratedby A / p , or equivalently, the hereditary torsion theory generated by the multiplicativelysubset A \ p . For every torsion theory σ we consider the following sets of ideals:(1) L ( σ ) , the Gabriel filter of σ .(2) Z ( σ ) = L ( σ ) ∩ Spec ( A ) . In particular, if p ⊆ q are prime ideals and p ∈ Z ( σ ) , then q ∈ Z ( σ ) .(3) C ( A , σ ) = { a | A / a ∈ F σ } .(4) K ( σ ) = C ( A , σ ) ∩ Spec ( A ) ; it is the complement of Z ( σ ) in Spec ( A ) . In particular,if p ⊆ q are prime ideals and q ∈ Z ( σ ) , then p ∈ Z ( σ ) .(5) C ( σ ) = Max K ( σ ) .If σ is of finite type, then σ = ∧{ σ A \ p | p ∈ K ( σ ) } . Otherwise, σ = ∧{ σ A \ p | p ∈ C ( σ ) } whenever A is σ –noetherian, because σ A \ q ≤ σ A \ p if p ⊆ q , for any prime ideals p , q .An A –module M is totally p –noetherian whenever M is totally σ A \ p –noetherian. Lemma 5.1.
Let A be a local (non necessarily noetherian) ring with maximal ideal m ,and M bean A –module. The followingstatementsare equivalent:15a) M isnoetherian.(b) M istotally σ A \ m –noetherian.P ROOF
It is immediate because every element in A \ m is invertible. ƒ Proposition 5.2.
Let σ a finite type hereditary torsion theory in A , and M be an A –module. The followingstatementsare equivalent:(a) M istotally σ –noetherian.(b) M istotally σ A \ p –noetherianforevery p ∈ C ( σ ) = Max K ( σ ) .P ROOF (a) ⇒ (b). It is immediate because σ ≤ σ A \ p .(b) ⇒ (a). Let N ⊆ M , for every m ∈ C ( σ ) there exists s m ∈ A \ m , and H m ⊆ N ,finitely generated, such that N s m ⊆ H m ⊆ N . Since b = P m s m A belongs to L ( σ ) = ∩ m L ( σ A \ m ) , there exists c ∈ L ( σ ) , finitely generated, such that c ⊆ b . In consequence,there are finitely many elements s m , . . . , s m t such that c ⊆ P ti = s m i A ⊆ b , and we have N c ⊆ P ti = H m i ⊆ N . Hence, N is totally σ –finitely generated, and M is totally σ –noetherian. ƒ When we take the hereditary torsion theory σ =
0, i.e, when L ( σ ) = { A } , we havethat M is noetherian if, and only if, M is totally σ A \ m –noetherian for every maximal ideal m ∈ Supp ( M ) , which is [
1, Proposition 12 ] . Let σ be a hereditary torsion theory in Mod − A , and a ⊆ A an ideal, then we have:(1) a is σ –principal if there exists a ∈ a such that Cl A σ ( aA ) = Cl A σ ( a ) .(2) A is a σ –principal ideal ring , σ -PIR, whenever every ideal is σ –principal.(3) A is a totally σ –principal ideal ring , totally σ –PIR, whenever every ideal is totally σ –principal. Proposition 6.1. (Kaplansky’s Theorem)
Let σ be a finite type hereditary torsionthe-oryin Mod − A , thefollowingstatementsare equivalent:(a) A istotally σ –PIR.(b) Everyprimeideal p ∈ K ( σ ) istotally σ –principal.See also [
1, Proposition 16 ] .P ROOF (a) ⇒ (b). Since for every p ∈ K ( σ ) , the result holds.(b) ⇒ (a). Let Γ = { a ⊆ A | a is not totally σ –principal } . If Γ = ∅ , since every chainin Γ has a upper bound in Γ , there exist maximal elements in Γ . If { a i } i ⊆ Γ is a chain,we take a = ∪ i a i . If a is not totally σ –principal, there are b ∈ L ( σ ) , finitely generated,16nd a ∈ a such that ab ⊆ aA ⊆ a , and there exists an index i such that a ∈ a i , hence a i b ⊆ ab ⊆ aA ⊆ a i , which is a contradiction.Let a ∈ Γ maximal; if Cl A σ ( a ) is totally σ –principal, then a is totally σ –principal, hence a = Cl A σ ( a ) is σ –closed.Let us show that a is prime. Let a , b ∈ A \ a such that a b ∈ a . Since a + aA is totally σ –principal, there exists x ∈ a + aA and b ∈ L ( σ ) , finitely generated, such that ( a + aA ) b ⊆ x A ⊆ a + aA . If x ∈ A , then b ⊆ ( a : a ) / ∈ L ( σ ) , which is a contradiction. If x / ∈ A , then b ⊆ ( a : x ) / ∈ L ( σ ) , which is a contradiction. In consequence, we can find a prime ideal a ∈ K ( σ ) which is not totally σ –principal which is a contradiction. ƒ Corollary 6.2.
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