A characterization of length-factorial Krull monoids
aa r X i v : . [ m a t h . A C ] J a n A CHARACTERIZATION OF LENGTH-FACTORIAL KRULL MONOIDS
ALFRED GEROLDINGER AND QINGHAI ZHONG
Abstract.
An atomic monoid is length-factorial if each two distinct factorizations of any element havedistinct factorization lengths. We provide a characterization of length-factorial Krull monoids in termsof their class groups and the distribution of prime divisors in the classes. Introduction and Main Results
By an atomic monoid, we mean a commutative unit-cancellative semigroup with identity in whichevery non-invertible element is a finite product of irreducible elements. The monoids we have in mindstem from ring and module theory. An atomic monoid H is said to be • half-factorial if for every element a ∈ H each two factorizations of a have the same length; • length-factorial if for every element a ∈ H each two distinct factorizations of a have distinctlengths.Thus, an atomic monoid is factorial if and only if it is half-factorial and length-factorial. A commutativering is said to be atomic (half-factorial resp. length-factorial) if its monoid of regular elements has therespective property. All these arithmetical properties can be characterized in terms of catenary degrees.Indeed, it is easy to verify that a monoid is factorial (half-factorial resp. length-factorial) if its catenarydegree c ( H ) = 0 (its adjacent catenary degree c adj ( H ) = 0 resp. its equal catenary degree c eq ( H ) = 0).Half-factoriality has been studied since the beginning of factorization theory and there is a huge amountof literature. Monotone and equal catenary degrees were first studied by Foroutan ([24]), and for somerecent contributions we refer to [40, 47, 27, 33, 30]. Length-factoriality was first studied (in differentterminology) by Coykendall and Smith ([15]), who showed that an atomic integral domain is length-factorial if and only if it is factorial. However, such a result is far from being true in the monoid case(we refer to recent contributions by Chapman, Coykendall, Gotti, and others [11, 37, 38, 14] as well asto work on monoids that are not length-factorial [12, 6]).In the present paper we focus on Krull monoids. Krull monoids are atomic and they are factorial ifand only if their class group is trivial. Let H be a Krull monoid with class group G and let G P ⊂ G denote the set of classes containing prime divisors. Then H is half-factorial if and only if the monoidof zero-sum sequences B ( G P ) over G P is half-factorial. There is a standing conjecture that for everyabelian group G ∗ there is a half-factorial Krull monoid (equivalently, a half-factorial Dedekind domain)with class group isomorphic to G ∗ ([35, Section 5]). The conjecture holds true for Warfield groups butnot even for finite cyclic groups G the structure or the maximal size of subsets G ⊂ G , for which B ( G )is half-factorial, are known in general ([48, 49]).Our main result provides a characterization of when a Krull monoid is length-factorial, in terms ofthe class group and the distribution of prime divisors in the classes. Recall that reduced Krull monoidsare uniquely determined by their class groups and by the distribution of prime divisors in the classes [28,Theorem 2.5.4]. Mathematics Subject Classification.
Theorem 1.1.
Let H be a Krull monoid. Then H = H × ×F ( P ) × H ∗ , where P is a set of representativesof prime elements of H , F ( P ) × H ∗ ∼ = H red , and H ∗ is a reduced Krull monoid without primes. Theclass groups C ( H ) of H and C ( H ∗ ) of H ∗ are isomorphic, and H is length-factorial if and only if H ∗ islength-factorial. Let G P ∗ ⊂ C ( H ∗ ) denote the set of classes containing prime divisors.Then H is length-factorial but not factorial if and only if every class of G P ∗ contains precisely one primedivisor, H ∗ ∼ = B ( G P ∗ ) , G P ∗ = { e , , . . . , e ,t , e , , . . . , e ,t , . . . , e k, , . . . , e k,t , g , . . . , g k , e , , . . . , e ,t , g } , and C ( H ∗ ) = h e , , . . . , e ,t , g i ⊕ . . . ⊕ h e k, , . . . , e k,t , g k i ∼ = ( Z t ⊕ Z /n Z ) k , where • t ∈ N , k, s , s , . . . , s t ∈ N with k + 1 = s + s + . . . + s t ≥ , independent elements e , , . . . , e ,t , e , , . . . , e ,t , . . . , e k, , . . . , e k,t ∈ C ( H ∗ ) of infinite order and independent elements g , . . . , g k ∈ C ( H ∗ ) , which are of infinite order in case t > and of finite order for t = 0 ; • s is the smallest integer such that s g i ∈ h e i, , . . . , e i,t i and − s g i = s e i, + . . . + s t e i,t for every i ∈ [1 , k ] ; • e ,j = − P ki =1 e i,j for all j ∈ [1 , t ] , g = − P ki =1 g i , and n = gcd( s , . . . , s t ) .Moreover, C ( H ∗ ) is a torsion group if and only if t = 0 and in that case we have C ( H ∗ ) ∼ = ( Z /n Z ) k ,where n ≥ , k ∈ N with k + 1 = n , and ord( g i ) = n for all i ∈ [1 , k ] . Theorem 1.1 shows in particular that, if H is a length-factorial Krull monoid, then H ∗ is finitelygenerated Krull with torsion-free quotient group, whence H ∗ is a normal affine monoid in the sense ofcombinatorial commutative algebra ([9]). We proceed with a series of corollaries. Based on the algebraiccharacterization of length-factorial Krull monoids given in Theorem 1.1, we start with the descriptionof their arithmetic. We explicitly determine the system L ( H ) of sets of lengths, which has been doneonly in seldom cases ([32]). In particular, the set of distances and the elasticity are finite (a geometriccharacterization of when the elasticity of Krull monoids with finitely generated class group are finite canbe found in [39]). Moreover, we observe that L ( H ) is additively closed, a quite rare property ([31]). Corollary 1.2 ( Arithmetic of length-factorial Krull monoids).
Let H be a length-factorial Krullmonoid, that is not factorial, and let all notation be as in Theorem 1.1. The inclusion B ( G P ∗ ) ֒ → F ( G P ∗ ) is a divisor theory with class group isomorphic to C ( H ) . Theset of atoms A ( G P ∗ ) = { U , . . . , U k , V , . . . , V t } where, for every i ∈ [0 , k ] and every j ∈ [1 , t ] , U = g s e s , · . . . · e s t ,t , U i = e s i, · . . . · e s t i,t , V = g · . . . · g k , V j = e ,j · . . . · e k,j and U · . . . · U k = V s · . . . · V s t t . Every B ∈ B ( G P ∗ ) can be written uniquely in the form B = ( U · . . . · U k ) x k Y i =0 U y i i t Y j =1 V z j j , where x, y , . . . , y k , z , . . . , z t ∈ N , y i = 0 for some i ∈ [0 , k ] , and z j < s j for some j ∈ [0 , t ] .Furthermore, we have L ( B ) = k X i =0 y i + t X j =0 z j + n ν ( k + 1) + ( x − ν ) t X j =0 s j : ν ∈ [0 , x ] o . For the system of sets of lengths L ( H ) , we have L ( H ) = n(cid:8) y + ν ( k + 1) + ( x − ν ) t X j =0 s j : ν ∈ [0 , x ] (cid:9) : y, x ∈ N o . In particular, the system L ( H ) is additively closed with respect to set addition as operation. CHARACTERIZATION OF LENGTH-FACTORIAL KRULL MONOIDS 3
Next we consider Krull monoids having some key properties, namely the approximation property or theproperty that every class contains at least one prime divisor. All Krull domains have the approximationproperty. Holomorphy rings in global fields are Dedekind domains with finite class group and infinitelymany prime divisors in all classes. Cluster algebras that are Krull ([25]) and monoid algebras that areKrull ([22]) are more recent examples of Krull domains having infinitely many prime divisors in all classes.Examples of Krull monoids stemming from module theory and having prime divisors in all classes will bediscussed in Section 2. Corollary 1.3 should be compared with the classical result that a Krull monoidhaving prime divisors in each class is half-factorial if and only if its class group has at most two elements.
Corollary 1.3.
Let H be a Krull monoid and H ∗ be as in Theorem 1.1. If H satisfies the approximation property, then H is length-factorial if and only if it is factorial. Suppose that every nonzero class of H contains a prime divisor. Then H is length-factorial if andonly if H ∗ ∼ = B ( C ( H ) \ { } ) and (cid:0) |C ( H ) | ≤ or C ( H ) is an elementary -group of rank two (cid:1) . As already said before, it was proved by Coykendall and Smith that a commutative integral domain islength-factorial if and only if it is factorial ([15]). Our next corollary shows that this result remains truefor commutative Krull rings with zero divisors and for normalizing (but not necessarily commutative)Krull rings.
Corollary 1.4 ( Length-factorial Krull rings). Let R be an additively regular Krull ring. Then R is length-factorial if and only if R is factorial. Let R be a normalizing Krull ring. Then R is length-factorial if and only if R is factorial. We end with a corollary on transfer Krull monoids. A monoid H is said to be transfer Krull if thereis a transfer homomorphism θ : H → B , where B is a Krull monoid. Thus, Krull monoids are transferKrull, with θ being the identity. However, in general, transfer Krull monoids need neither be cancellativenor completely integrally closed nor v -noetherian. We discuss an example after the proof of Corollary 1.5(Example 2.7) and refer to the survey [34] for more. In particular, all half-factorial monoids are transferKrull but not necessarily Krull. But reduced length-factorial transfer Krull monoids are Krull, as weshow in our final corollary. Corollary 1.5 ( Length-factorial transfer Krull monoids).
Let H be a transfer Krull monoid. If H is length-factorial, then H red is Krull whence it fulfills the structural description given in Theorem 1.1. All results of the present paper, as well as prior work done in [11], indicate that length-factoriality is amuch more exceptional property than half-factoriality and that this is true not only for domains (whichis known since [15]) but also for commutative and cancellative monoids. The innocent Example 2.2 seemsto suggest that the situation is quite different for commutative semigroups that are unit-cancellative butnot necessarily cancellative.2.
Proof of Theorem 1.1 and of its corollaries
Our notation and terminology are consistent with [28]. We gather some key notions. For every positiveinteger n ∈ N , C n denotes a cyclic group with n elements. For integers a, b ∈ Z , [ a, b ] = { x ∈ Z : a ≤ x ≤ b } denotes the discrete interval between a and b . For subsets A, B ⊂ Z , A + B = { a + b : a ∈ A, b ∈ B } denotes their sumset and the set of distances ∆( A ) ⊂ N is the set of all d ∈ N for which there is an element a ∈ A such that [ a, a + d ] ∩ A = { a, a + d } . For a set L ⊂ N , we let ρ ( L ) = sup L/ min L ∈ Q ≥ ∪ {∞} denote the elasticity of L , and we set ρ ( { } ) = 1.Let H be a commutative semigroup with identity. We denote by H × the group of invertible elements.We say that H is reduced if H × = { } and we denote by H red = { aH × : a ∈ H } the associated reducedsemigroup. An element u ∈ H is said to be cancellative if au = bu implies that a = b for all a, b, u ∈ H . ALFRED GEROLDINGER AND QINGHAI ZHONG
The semigroup H is called • cancellative if all elements of H are cancellative; • unit-cancellative if a, u ∈ H and a = au implies that u ∈ H × .Thus, every cancellative monoid is unit-cancellative. Throughout this paper, a monoid means a commutative and unit-cancellative semigroup with identity.
For a set P , let F ( P ) be the free abelian monoid with basis P . An element a ∈ F ( P ) is written in theform a = Y p ∈ P p v p ( a ) ∈ F ( P ) , where v p : F ( P ) → N denotes the p -adic valuation. Then | a | = P p ∈ P v p ( a ) ∈ N is the length of a andsupp( a ) = { p ∈ P : v p ( a ) > } ⊂ P is the support of a . Let H be a multiplicatively written monoid. Anelement u ∈ H is said to be • prime if u / ∈ H × and, for all a, b ∈ H with u | ab , u ∤ a implies u | b . • irreducible (or an atom ) if u / ∈ H × and, for all a, b ∈ H , u = ab implies that a ∈ H × or b ∈ H × .We denote by A ( H ) the set of atoms of H and, if H is cancellative, then q ( H ) is the quotient group of H . The free abelian monoid Z ( H ) = F ( A ( H red )) is the factorization monoid of H and π : Z ( H ) → H red ,defined by π ( u ) = u for all u ∈ A ( H red ), is the factorization homomorphism of H . For an element a ∈ H , • Z H ( a ) = Z ( a ) = π − ( aH × ) ⊂ Z ( H ) is the set of factorizations of a , and • L H ( a ) = L ( a ) = {| z | : z ∈ Z ( a ) } is the set of lengths of a .Note that L ( a ) = { } if and only if a ∈ H × . Then H is atomic (resp. factorial) if Z ( a ) = ∅ (resp. | Z ( a ) | = 1) for all a ∈ H . Examples of atomic monoids, that are not necessarily cancellative, includesemigroups of ideals and semigroups of isomorphism classes of modules (see [23, Section 3.2 and 3.3], [30,Section 4], and Examples 2.2 and 2.7). If H is atomic, then1 ≤ | L ( a ) | ≤ | Z ( a ) | for all a ∈ H ,
We say that the monoid H is • half-factorial if 1 = | L ( a ) | for all a ∈ H , and • length-factorial if 1 ≤ | L ( a ) | = | Z ( a ) | for all a ∈ H .Thus, by definition, H is factorial if and only if it is half-factorial and length-factorial. Furthermore, H is factorial (half-factorial resp. length-factorial) if and only if H red has the respective property. Then L ( H ) = { L ( a ) : a ∈ H } is the system of sets of lengths of H , ∆( H ) = [ L ∈L ( H ) ∆( L ) ⊂ N is the set of distances of H , and ρ ( H ) = sup { ρ ( L ) : L ∈ L ( H ) } ∈ R ≥ ∪ {∞} is the elasticity of H . We say that H has accepted elasticity if there is L ∈ L ( H ) such that ρ ( L ) = ρ ( H ).If H is not half-factorial, then min ∆( H ) = gcd ∆( H ). We start with a simple lemma. Lemma 2.1.
Let H be a length-factorial monoid. ρ ( H ) < ∞ . If H is cancellative, then the elasticity ρ ( H ) is accepted. If H is cancellative but not factorial, then | ∆( H ) | = 1 . CHARACTERIZATION OF LENGTH-FACTORIAL KRULL MONOIDS 5
Proof.
Without restriction we may suppose that H is reduced. By definition, H is half-factorial if andonly if ρ ( H ) = 1 if and only if ∆( H ) = ∅ , and if this holds, then the elasticity is accepted. Thus we maysuppose that H is not half-factorial.1. Assume to the contrary that ρ ( H ) is infinite and choose an element a ∈ H with ρ ( L ( a )) >
1. Thenthere exist r, s, t ∈ N and u , . . . , u r , v , . . . , v s , w , . . . , w t ∈ A ( H ) with { v , . . . , v s } ∩ { w , . . . , w t } = ∅ such that a = u · . . . · u r v · . . . · v s = u · . . . · u r w · . . . · w t with ρ ( L ( a )) = ( r + t ) / ( r + s ) > . Since ρ ( H ) is infinite, there exists b ∈ H such that ρ ( L ( b )) > t/s . Moreover, there exist r ′ , s ′ , t ′ ∈ N and x , . . . , x r ′ , y , . . . , y s ′ , z , . . . , z t ′ ∈ A ( H ) with { y , . . . , y s ′ } ∩ { z , . . . , z t ′ } = ∅ such that b = x · . . . · x r ′ y · . . . · y s ′ = x · . . . · x r ′ z · . . . · z t ′ with ρ ( L ( b )) = ( r ′ + t ′ ) / ( r ′ + s ′ ) > t/s . Since a t ′ − s ′ b t − s = ( u · . . . · u r v · . . . · v s ) t ′ − s ′ ( x · . . . · x r ′ z · . . . · z t ′ ) t − s = ( x · . . . · x r ′ y · . . . · y s ′ ) t − s ( u · . . . · u r w · . . . · w t ) t ′ − s ′ and since H is length-factorial, we obtain that( u · . . . · u r v · . . . · v s ) t ′ − s ′ ( x · . . . · x r ′ z · . . . · z t ′ ) t − s and ( x · . . . · x r ′ y · . . . · y s ′ ) t − s ( u · . . . · u r w · . . . · w t ) t ′ − s ′ are equal in the factorization monoid Z ( H ). Since { v , . . . , v s } ∩ { w , . . . , w t } = ∅ and { y , . . . , y s ′ } ∩ { z , . . . , z t ′ } = ∅ , it follows that ( v · . . . · v s ) t ′ − s ′ and ( y · . . . · y s ′ ) t − s are equal in the factorization monoid Z ( H ), whence s ( t ′ − s ′ ) = s ′ ( t − s ). Therefore t/s = t ′ /s ′ > ρ ( L ( b )), a contradiction.2. This proof runs along similar lines as the proof of the first assertion. But, we need to usecancellativity now which is not needed in 1. (see Example 2.2). Assume to the contrary that ρ ( H )is not accepted and choose an element a ∈ H with ρ ( L ( a )) >
1. Then there exist r, s, t ∈ N and u , . . . , u r , v , . . . , v s , w , . . . , w t ∈ A ( H ) with { v , . . . , v s } ∩ { w , . . . , w t } = ∅ such that a = u · . . . · u r v · . . . · v s = u · . . . · u r w · . . . · w t with ρ ( L ( a )) = ( r + t ) / ( r + s ) > . Let a = v · . . . · v s . Then ρ ( L ( a )) = t/s >
1. Since ρ ( H ) is not accepted, there exists b ∈ H suchthat ρ ( L ( b )) > ρ ( L ( a )). Moreover, there exist r ′ , s ′ , t ′ ∈ N and x , . . . , x r ′ , y , . . . , y s ′ , z , . . . , z t ′ ∈ A ( H )with { y , . . . , y s ′ } ∩ { z , . . . , z t ′ } = ∅ such that b = x · . . . · x r ′ y · . . . · y s ′ = x · . . . · x r ′ z · . . . · z t ′ with ρ ( L ( b )) = ( r ′ + t ′ ) / ( r ′ + s ′ ) > ρ ( a ) . Let b = y · . . . · y s ′ . Then ρ ( L ( b )) = t ′ /s ′ > ρ ( a ). Since H is length-factorial and a t ′ − s ′ b t − s = ( v · . . . · v s ) t ′ − s ′ ( z · . . . · z t ′ ) t − s = ( y · . . . · y s ′ ) t − s ( w · . . . · w t ) t ′ − s ′ , it follows from { v , . . . , v s } ∩ { w , . . . , w t } = ∅ and { y , . . . , y s ′ } ∩ { z , . . . , z t ′ } = ∅ that ( v · . . . · v s ) t ′ − s ′ and ( y · . . . · y s ′ ) t − s are equal in the factorization monoid Z ( H ), whence s ( t ′ − s ′ ) = s ′ ( t − s ). Therefore,we infer that ρ ( L ( a )) = t/s = t ′ /s ′ = ρ ( L ( b )), a contradiction.3. Assume to the contrary that | ∆( H ) | ≥
2. Since min ∆( H ) = gcd ∆( H ), we may choose d, d ∈ ∆( H )with d = d such that d divides d . Let r, s, k, t ∈ N and u , . . . , u r , v , . . . , v s , w , . . . , w s + d , x , . . . , x k , y , . . . , y t , z , . . . , z t + d ∈ A ( H )with { v , . . . , v s } ∩ { w , . . . , w s + d } = ∅ and { y , . . . , y t } ∩ { z , . . . , z t + d } = ∅ such that a = u · . . . · u r v · . . . · v s = u · . . . · u r w · . . . · w s + d , with L ( a ) ∩ [ r + s, r + s + d ] = { r + s, r + s + d } ,b = x · . . . · x k y · . . . · y t = x · . . . · x k z · . . . · z t + d , with L ( b ) ∩ [ k + t, k + t + d ] = { k + t, k + t + d } . ALFRED GEROLDINGER AND QINGHAI ZHONG
Then a d b d = ( u · . . . · u r v · . . . · v s ) d ( x · . . . · x k z · . . . · z t + d ) d = ( x · . . . · x k y · . . . · y t ) d ( u · . . . · u r w · . . . · w s + d ) d . Since d ( r + s ) + d ( k + t + d ) = d ( k + t ) + d ( r + s + d ) and H is length-factorial, we obtain that thetwo factorizations ( v · . . . · v s ) d ( z · . . . · z t + d ) d and ( y · . . . · y t ) d ( w · . . . · w s + d ) d are equal (in thefactorization monoid Z ( H )). Since { v , . . . , v s } ∩ { w , . . . , w s + d } = ∅ , we obtain ( v · . . . · v s ) d divides( y · . . . · y t ) d in Z ( H ). Since { y , . . . , y t } ∩{ z , . . . , z t + d } = ∅ , we obtain ( y · . . . · y t ) d divides ( v · . . . · v s ) d in Z ( H ), whence ( v · . . . · v s ) d = ( y · . . . · y t ) d ∈ Z ( H ). It follows that y · . . . · y t = ( v · . . . · v s ) d/d andhence b = x · . . . · x k ( v · . . . · v s ) d/d = x · . . . · x k ( v · . . . · v s ) d/d − w · . . . · w s + d , which implies that k + t + d ∈ L ( b ) ∩ [ k + t, k + t + d ], a contradiction. (cid:3) Our next example shows that the elasticity of a non-cancellative length-factorial monoid does not needto be accepted and that the set of distances may contain more than one element.
Example 2.2.
1. Let R be a ring and C be a small class of left R -modules that is closed under finite direct sums,direct summands, and isomorphisms. Then the set V ( C ) of isomorphism classes of modules from C isa reduced commutative semigroup, with operation induced by the direct sum ([5]). Suppose that allmodules from C are directly finite (or Dedekind finite), which means thatIf M, N are modules from C such that M ∼ = M ⊕ N , then N = 0.This property holds true for large classes of modules (including all finitely generated modules over com-mutative rings; for more see [36, 19]) and is equivalent to V ( C ) being unit-cancellative. We will meet suchmonoids V ( C ) at several places of the manuscript (e.g., in Example 2.7).2. For m ∈ N , let us consider the commutative monoid H m generated by A m = { a , . . . , a m , u , u } with relations generated by R m = { ( a u , a u ) , ( a u , a u ) , . . . , ( a m u m , a m u m ) } , say H m = h a , . . . , a m , u , u | a u = a u , a u = a u , . . . , a m u m = a m u m i . Then H m is a reduced, commutative, atomic, non-cancellative monoid with A ( H m ) = A m . By con-struction, we have [1 , m ] ⊂ ∆( H m ), ρ ( H m ) = 3 /
2, and ρ ( H ) is not accepted. We assert that H m islength-factorial.We define, for any a, b ∈ H m , that a ∼ b if there exists c ∈ H m such that ac = bc . This is a congruencerelation on H m and the monoid H m, canc = H m / ∼ is the associated cancellative monoid of H m . Forevery a ∈ H m , we denote by [ a ] ∈ H m, canc the congruence class of H . Then H m, canc ∼ = F ( { [ a i ] : i ∈ [1 , m ] } ) × h [ u ] , [ u ] | [ u ] = [ u ] i , whence it is easy to see that H m, canc is length-factorial. Let x , x be two atoms of H m . By ourconstruction of H , we have [ x ] = [ x ] if and only if x = x . Therefore the length-factoriality of H m, canc implies that H m is length-factorial. By a result of Bergman-Dicks ([7, Theorems 6.2 and 6.4] and [8, page315]), the monoid H m can be realized as a monoid of isomorphism classes of modules, as introduced in 1.Next we discuss Krull monoids. A monoid homomorphism ϕ : H → D is called a • divisor homomorphism if a, b ∈ H and ϕ ( a ) | ϕ ( b ) (in D ) imply that a | b (in H ); • divisor theory (for H ) if ϕ is a divisor homomorphism, D is free abelian, and for every a ∈ D thereare a , . . . , a m ∈ H such that a = gcd (cid:0) ϕ ( a ) , . . . , ϕ ( a m ) (cid:1) .A monoid H is a Krull monoid if it is cancellative and satisfies one of the following equivalent conditions([28, Theorem 2.4.8] ):(a) H is completely integrally closed and satisfies the ACC on divisorial ideals.(b) H has a divisor homomorphism to a free abelian monoid.(c) H has a divisor theory. CHARACTERIZATION OF LENGTH-FACTORIAL KRULL MONOIDS 7
Property (a) can be used to show that a domain is a Krull domain if and only if its multiplicative monoidof nonzero elements is a Krull monoid. Examples of Krull monoids are given in [28] and in the recentsurvey [34]). In particular, let V ( C ) be a monoid of isomorphism classes of modules, as introduced inExample 2.2.1. If End R ( M ) is semilocal for all M from C , then V ( C ) is a reduced Krull monoid ([16,Theorem 3.4]), and every reduced Krull monoid can be realized as a monoid of isomorphism classes ofmodules ([21, Theorem 2.1]).To discuss class groups of Krull monoids, let H be a Krull monoid. Then there is a divisor theory H red ֒ → F = F ( P ) and(2.1) C ( H ) = C ( H red ) = q ( F ) / q ( H red )is the (divisor) class group of H . The divisor class group is isomorphic to the (ideal theoretic) v -classgroup of H , and if R is a Krull domain, then the class group of the Krull monoid R \ { } coincides withthe usual divisor class group of the domain R . If the monoid H in Theorem 1.1 is length-factorial, then H ∗ is a reduced finitely generated Krull monoid. There are various characterizations of finitely generatedKrull monoids ([28, Theorem 2.7.14]). In particular, every such monoid is a Diophantine monoid (themonoid of non-negative solutions of a system of linear Diophantine equations; [13]). For every a ∈ q ( F ),we denote by [ a ] = a q ( H red ) ⊂ q ( F ) the class containing a . For g ∈ C ( H ), P ∩ g is the set of primedivisors lying in g . Concerning the distribution of prime divisors in Krull monoids of isomorphism classesof modules we refer to [20, 17, 41, 18, 2].Let G be an additive abelian group and G ⊂ G be a subset. We denote by h G i ⊂ G the subgroupgenerated by G and by [ G ] ⊂ G the submonoid generated by G . A tuple ( e , . . . , e r ) ∈ G r , with r ∈ N (respectively, the elements e , . . . , e r ∈ G ) are called independent if e i = 0 for all i ∈ [1 , r ] and h e , . . . , e r i = h e i⊕ . . . ⊕h e r i , and it is called a basis of G if e i = 0 for all i ∈ [1 , r ] and G = h e i⊕ . . . ⊕h e r i .We discuss a class of Krull monoids needed in the sequel, namely monoids of zero-sum sequences. Foran element S = g · . . . · g ℓ = Y g ∈ G g v g ( S ) ∈ F ( G ) , where g , . . . , g ℓ ∈ G , | S | = ℓ = P g ∈ G v g ( S ) ∈ N is the length of S , and σ ( S ) = g + . . . + g ℓ ∈ G is the sum of S .
We say that S is zero-sum free if P i ∈ I g i = 0 for all ∅ 6 = I ⊂ [1 , ℓ ]. The monoid of zero-sum sequences B ( G ) = { S ∈ F ( G ) : σ ( S ) = 0 } ⊂ F ( G )over G is a Krull monoid, by Property (b), since the inclusion B ( G ) ֒ → F ( G ) is a divisor homomor-phism. We denote by A ( G ) := A (cid:0) B ( G ) (cid:1) the set of atoms (minimal zero-sum sequences) of B ( G ).The subset G is called half-factorial (non-half-factorial resp. minimal non-half-factorial) if the monoid B ( G ) is half-factorial (not half-factorial resp. G is not half-factorial but every proper subset is half-factorial). Half-factorial and (minimal) non-half-factorial subsets play a central role when studying thearithmetic of Krull monoids (we refer to [28, Chapter 6] for the basics and to [54, 50]). Note that minimalnon-half-factorial subsets are finite.The arithmetic of Krull monoids is studied via transfer homomorphisms to monoids of zero-sum se-quences. We recall the required concepts. A monoid homomorphism θ : H → B is called a transferhomomorphism if it has the following properties: (T 1) B = θ ( H ) B × and θ − ( B × ) = H × . (T 2) If u ∈ H , b, c ∈ B and θ ( u ) = bc , then there exist v, w ∈ H such that u = vw , θ ( v ) ∈ bB × ,and θ ( w ) ∈ cB × . Lemma 2.3.
Let θ : H → B be a transfer homomorphism of atomic monoids. For every a ∈ H , we have L H ( a ) = L B (cid:0) θ ( a ) (cid:1) . ALFRED GEROLDINGER AND QINGHAI ZHONG Let p ∈ H . Then p is an atom in H if and only if θ ( p ) is an atom in B . Moreover, if p is a primein H , then θ ( p ) is a prime in B . L ( H ) = L ( B ) , whence H is half-factorial if and only if B is half-factorial. If H is length-factorial, then B is length-factorial.Proof. Without restriction we may suppose that H and B are reduced. Then (T1) implies that θ issurjective.1. This easily follows from (T 2) (for details in the cancellative setting we refer to [28, Chapter 3.2]).2. Let p ∈ H . Since p is an atom in H if and only if L H ( p ) = { } and similarly for θ ( p ) and B , 1.implies that p is an atom in H if and only if θ ( p ) is an atom in B .Now suppose that p is a prime in H and let α, β ∈ B such that θ ( p ) | αβ . Then there is c ∈ H suchthat αβ = θ ( pc ). Then (T2) implies that there are a, b ∈ H such that pc = ab , θ ( a ) = α , and θ ( b ) = β .Without restriction we may suppose that p | a , say a = pa ′ for some a ′ ∈ H , whence α = θ ( a ) = θ ( p ) θ ( a ′ ).Thus θ ( p ) is a prime in B .3. This follows immediately from 1.4. Suppose that H is length-factorial and choose some α ∈ B . Let a ∈ H such that θ ( a ) = α , and let k ∈ L B ( α ) = L H ( a ). By (T2) , every factorization of α of length k can be lifted to a factorization of a oflength k . Thus, if there is only one factorization of a of length k , there is only one factorization of α oflength k . This implies that B is length-factorial. (cid:3) Let all notation be as in Lemma 2.3. There are examples (even for cancellative monoids) where θ ( p )is a prime in B but p fails to be prime in H . Furthermore, B may be length-factorial, but H is notlength-factorial. Lemma 2.4.
Let H be a reduced Krull monoid with divisor theory H ֒ → F = F ( P ) , class group G = C ( H ) , and let G P = { [ p ] : p ∈ P } ⊂ G be the set of classes containing prime divisors. The map β : H → B ( G P ) , defined by a = p · . . . · p ℓ [ p ] · . . . · [ p ℓ ] where ℓ ∈ N and p , . . . , p ℓ ∈ P ,is a transfer homomorphism. The map β is an isomorphism if and only if every class g ∈ G P contains precisely one primedivisor. We have G = [ G P ] and G = [ G P \ { g } ] for all classes g ∈ G P that contain precisely one primedivisor.Proof.
1. This follows from [28, Theorem 3.4.10].2. Since B ( G P ) is reduced, (T 1) implies that β is surjective. Thus, β is an isomorphism if and onlyif every class g ∈ G P contains precisely one prime divisor.3. This follows from [28, Theorem 2.5.4]. (cid:3) Lemma 2.5.
Let G be an abelian group and let G ⊂ G \ { } be a subset such that G = [ G \ { g } ] forall g ∈ G . Suppose there is B ∈ B ( G ) having two distinct factorizations B = U · . . . · U k = V · . . . · V ℓ , where k, ℓ ≥ and U , . . . , U k , V , . . . , V ℓ ∈ A ( G ) . For any distinct g, h ∈ G , there exists two atoms A , A ∈ A ( G ) such that v g ( A ) = 1 and h ∈ supp( A ) ⊂ G \ { g } . If B ( G ) is length-factorial, then A ( G ) = { U , . . . , U k , V , . . . , V ℓ } .Proof.
1. Let g, h ∈ G with g = h . Since − h ∈ G = [ G \ { g } ], there is an atom A ∈ A ( G \ { g } )such that h ∈ supp( A ) ⊂ G \ { g } . Since − g ∈ G = [ G \ { g } ], there is an atom A ∈ A ( G ) such that v g ( A ) = 1.2. Suppose B ( G ) is length-factorial. Assume to the contrary there is an atom A ∈ A ( G ) \{ U , . . . , U k , V , . . . , V ℓ } . If | supp( A ) | = 1, say supp( A ) = { g } , then ord( g ) is finite and by 1. there CHARACTERIZATION OF LENGTH-FACTORIAL KRULL MONOIDS 9 exists an atom A with v g ( A ) = 1, whence A = A . Therefore A divides A ord( g )1 . If | supp( A ) | ≥
2, thenfor every g ∈ supp( A ), it follows by 1. that there exists an atom A g ∈ A ( G ) with g ∈ supp( A g ) suchthat supp( A ) supp( A g ). Then A = A g for every g ∈ supp( A ) and A divides Q g ∈ supp( A ) A v g ( A ) g .To sum up, there exist s ∈ N and atoms W , . . . , W s with A = W i for every i ∈ [1 , s ] such that A divides W · . . . · W s . We may suppose W · . . . · W s = AX · . . . · X t , where t ≥ X , . . . , X t ∈ A ( G ).If ℓ = k or t = s , then B ( G ) is not length-factorial, a contradiction. Suppose ℓ = k and t = s . Bysymmetry, we may suppose that ℓ > k . If t > s , then( W · . . . · W s ) ℓ − k ( V · . . . · V ℓ ) t − s = ( U · . . . · U k ) t − s ( AX · . . . · X t ) ℓ − k has two distinct factorizations of length ℓt − sk , whence B ( G ) is not length-factorial, a contradiction. If s > t , then ( W · . . . · W s ) ℓ − k ( U · . . . · U k ) s − t = ( V · . . . · V ℓ ) s − t ( AX · . . . · X t ) ℓ − k has two distinct factorizations of length sℓ − tk , whence B ( G ) is not length-factorial, a contradiction. (cid:3) Lemma 2.6.
Let G be an abelian group and let G ⊂ G \ { } be a subset such that [ G \ { g } ] = G forall g ∈ G . Suppose that B ( G ) is length-factorial but not factorial. G is a minimal non-half-factorial set. For every g ∈ G , there exist A ∈ A ( G ) such that v g ( A ) = 1 and |{ A ∈ A ( G ) : v g ( A ) > }| = 2 . For any two distinct atoms A , A ∈ A ( G ) , either supp( A ) ∩ supp( A ) = ∅ or | gcd( A , A ) | = 1 .Proof. Since B ( G ) is length-factorial but not factorial, it is not half-factorial.1. There is a B ∈ B ( G ) such that | L ( B ) | ≥
2, which implies that supp( B ) is not half-factorial. Let G ⊂ supp( B ) be a minimal non-half-factorial subset and let B ∈ B ( G ) such that | L ( B ) | ≥
2. ThenLemma 2.5.2 implies A ( G ) = A ( G ).Assume to the contrary that G \ G = ∅ . Let h ∈ G \ G . Then by Lemma 2.5.1 there is an atom A ∈ A ( G ) with h ∈ supp( A ), whence A
6∈ A ( G ), a contradiction. Therefore G = G is a minimalnon-half-factorial subset.2. Let g ∈ G . By Lemma 2.5.1, there exists an atom A such that v g ( A ) = 1 and hence | supp( A ) | ≥
2. Let h ∈ supp( A ) \ { g } . Then Lemma 2.5.1 implies there exists an atom A g ∈ B ( G \ { h } ) suchthat g ∈ supp( A g ). Thus A g = A . Furthermore, for every h ∈ supp( A ) \ { g } , Lemma 2.5.1 implies thatthere exists an atom A h ∈ B ( G \ { g } ) such that h ∈ supp( A h ).Assume to the contrary that there exists an atom A ∈ A ( G ) \ { A , A g } such that g ∈ supp( A ).Therefore A Y h ∈ G \{ g } A v g ( A ) v h ( A ) h = A v g ( A )1 X · . . . · X s , where s ∈ N and X , . . . , X s ∈ A ( G \ { g } ). It follows by Lemma 2.5.2 that A g ∈ { A , A } ∪ { A h : h ∈ supp( A ) \ { g }} ∪ { X i : i ∈ [1 , s ] } , a contradiction.3. Let A , A ∈ A ( G ) be distinct such that supp( A ) ∩ supp( A ) = ∅ . Assume to the contrarythat there are g, h ∈ G such that gh divides gcd( A , A ). By 2., there is no other atom A such thatsupp( A ) ∩ { g, h } 6 = ∅ . If g = h , then there is no atom A with v g ( A ) = 1, a contradiction to 2. If g = h ,then − h ∈ [ G \{ g } ] implies that there is an atom A ∈ A ( G \{ g } ) with h ∈ supp( A ), a contradiction. (cid:3) Proof of Theorem 1.1.
Let H be a Krull monoid. By [28, Theorem 2.4.8], there is a decomposition H = H × × H , where H is a reduced Krull monoid, isomorphic to H red . If P ⊂ H is the set of primeelements of H and H ∗ = { a ∈ H : p ∤ a for all p ∈ P } , then H = F ( P ) × H ∗ ([28, Theorem 1.2.3]).Clearly, H ∗ is a reduced Krull monoid. By definition, H is length-factorial if and only if H red ∼ = H islength-factorial, and H is length-factorial if and only if H ∗ is length-factorial. Let H ∗ ֒ → F ( P ∗ ) be a divisor theory. Then H = F ( P ) × H ∗ ֒ → F ( P ) × F ( P ∗ ) = F ( P ), where P = P ⊎ P ∗ , is a divisor theory, whence we obtain that (we use (2.1)) C ( H ) = C ( H ) = q ( F ( P )) / q ( H )= q ( F ( P )) × q ( F ( P ∗ )) / q ( F ( P )) × q ( H ∗ ) ∼ = q ( F ( P ∗ )) / q ( H ∗ ) = C ( H ∗ ) . Let G P ∗ ⊂ C ( H ∗ ) denote the set of classes containing prime divisors, and note that 0 G P ∗ . It remainsto prove the characterization of length-factoriality. Note that the Moreover statement, dealing with thecase of torsion class groups, follows immediately from the main statement. We proceed in two steps.
Step 1.
Suppose that H and H ∗ are length-factorial but not factorial.Assume to the contrary that there exist distinct p, q ∈ P ∗ such that 0 = [ p ] = [ q ] ∈ C ( H ∗ ). Since H ∗ ֒ → F ( P ∗ ) is a divisor theory, there exist r ≥ a , . . . , a r ∈ H ∗ such that p = gcd( a , . . . , a r ). Without loss of generality, we may assume that a , . . . , a r ∈ A ( H ∗ ).Let a = p k q · . . . · q s p · . . . · p ℓ , where k ≥ s ≥ ℓ ≥ q , . . . , q s ∈ P ∗ \ { p } with [ q j ] = [ p ] for j ∈ [1 , s ], and p , . . . , p ℓ ∈ P ∗ with [ p i ] = [ p ] for i ∈ [2 , ℓ ]. If k + s ≥
2, then b = p k + s p · . . . · p ℓ and b = q k + s p · . . . · p ℓ are both atoms of H ∗ . We observe that b b = ( p k + s − qp · . . . · p ℓ )( pq k + s − p · . . . · p ℓ )has two distinct factorizations of length two, a contradiction. Thus k + s = 1 and a = pp · . . . · p ℓ .Similarly, we may assume that a = pp ′ · . . . · p ′ ℓ ′ , where ℓ ′ ≥ p ′ i ] = [ p ] for i ∈ [2 , ℓ ′ ]. We observethat a ( qp ′ · . . . · p ′ ℓ ′ ) = ( qp · . . . · p ℓ ) a has two distinct factorizations of length two, a contradiction. Therefore, every nonzero class g ∈ C ( H ∗ )contains at most one prime divisor. Thus, Lemma 2.4.2 implies that β : H ∗ → B ( G P ∗ ) is an isomorphism,whence H ∗ ∼ = B ( G P ∗ ) and B ( G P ∗ ) is length-factorial but not factorial.It remains to determine the structure of G P ∗ . Since H ∗ ֒ → F ( P ∗ ) is a divisor theory and every classof C ( H ∗ ) contains at most one prime divisor, we obtain that C ( H ∗ ) = [ G P ∗ \ { g } ] for all g ∈ G P ∗ byLemma 2.4.3. Thus, the assumption of Lemma 2.6 is satisfied which implies that G P ∗ is a minimalnon-half-factorial set. Let B ∈ B ( G P ∗ ) with | L ( B ) | ≥ | B | be minimal with this property, say B = U U · . . . · U k = V V · . . . · V ℓ , where k, ℓ ∈ N with k = ℓ , and U , U , . . . , U k , V , V , . . . , V ℓ ∈ A ( G P ∗ ). Then Lemma 2.5.2 implies that { U , U , . . . , U k , V , V , . . . , V ℓ } = A ( G P ∗ ) . The minimality of | B | implies that U i = V j for every i ∈ [0 , k ] and every j ∈ [0 , ℓ ]. If there exist j ∈ [0 , ℓ ]and a proper subset I ( [0 , k ] such that V j divides Q i ∈ I U i , then Q i ∈ I U i has two distinct factorizations, acontradiction to either the minimality of | B | or the length-factoriality of B ( G P ∗ ). Therefore, gcd( U i , V j ) =1 ∈ F ( G P ∗ ) for every i ∈ [0 , k ] and j ∈ [0 , ℓ ], whence | gcd( U i , V j ) | = 1 by Lemma 2.6.3. It follows that | U i | = ℓ + 1 and | V j | = k + 1 for every i ∈ [0 , k ] and j ∈ [0 , ℓ ]. Since | gcd( Y i ∈ I U i , B ) | = | I | ( ℓ + 1) , | gcd( Y i ∈ I U i , Y j ∈ J V j ) | ≤ | I || J | , and | gcd( Y i ∈ I U i , Y j ∈ [0 ,ℓ ] \ J V j ) | ≤ | I | ( ℓ + 1 − | J | )for every I ⊂ [0 , k ] and every J ⊂ [0 , ℓ ], we obtain that(2.2) | gcd( Y i ∈ I U i , Y j ∈ J V j ) | = | I || J | . For every g ∈ G P ∗ , there exist i ∈ [0 , k ] and j ∈ [0 , ℓ ] such that g ∈ supp( U i ) ∩ supp( U j ). Then, by Lemma2.6.2, for any i , i ∈ [0 , k ] and any j , j ∈ [0 , ℓ ] we have either U i = U i or supp( U i ) ∩ supp( U i ) = ∅ and either V j = V j or supp( V j ) ∩ supp( V j ) = ∅ . CHARACTERIZATION OF LENGTH-FACTORIAL KRULL MONOIDS 11
Assume to the contrary that there exist distinct i , i ∈ [0 , k ] and distinct j , j ∈ [0 , ℓ ] such that U i = U i and V j = V j . Then gcd( U i , V j ) = g for some g ∈ G P ∗ and hence gcd( U i U i , V j V j ) = g ,a contradiction to Equation (2.2). Thus, by symmetry, we may suppose U i = U i for any distinct i , i ∈ [0 , k ]. Therefore supp( U i ) ∩ supp( U i ) = ∅ for all distinct i , i ∈ [0 , k ]. Assume to the contrarythat there exist g ∈ G P ∗ and j ∈ [0 , ℓ ] such that v g ( V j ) ≥
2. Then there is i ∈ [0 , k ] such that v g ( U i ) ≥ A ∈ A ( G P ∗ ) with v g ( A ) = 1, a contradiction to Lemma 2.6.2. Thus v g ( V j ) = 1for all g ∈ supp( V j ) and all j ∈ [0 , ℓ ].We set U = g s e s , · . . . · e s t ,t , where s , . . . , s t ∈ N and g , e , , . . . , e ,t ∈ G P ∗ are pairwise distinct.After renumbering if necessary, we may suppose e ,i ∈ supp( V i ) for every i ∈ [1 , t ] and g ∈ supp( V ).Note that if supp( V j ) ∩ supp( V j ) = ∅ , then V j = V j , where j , j ∈ [0 , ℓ ]. Therefore, B = U · . . . · U k = V · . . . · V ℓ = V s V s · . . . · V s t t . The length-factoriality of B ( G P ∗ ) implies that k + 1 = s + . . . + s t . Since supp( U i ) ∩ supp( U i ) = ∅ forany two distinct i , i ∈ [0 , k ], U , . . . , U k and V , . . . , V t can be written as the form U i = g s i e s i, · . . . · e s t i,t , V j = e ,j e ,j · . . . · e k,j and U = g s e s , . . . e s t ,t , and V = g g · . . . · g k , where e , , . . . , e ,t , . . . , e k, , . . . , e k,t , g , . . . , g k ∈ G P ∗ , g = − P ki =1 g i , and e ,j = − P ki =1 e i,j for every j ∈ [1 , t ]. For each i ∈ [0 , k ], B ( { g i , e i, , . . . , e i,t } ) is half-factorial and length-factorial, whence it isfactorial and A ( { g i , e i, , . . . , e i,t } ) = { U i } . Thus, we obtain that s is the minimal integer such that − s g i ∈ h e i, , . . . , e i,t i .In order to show that ( e , , . . . , e ,t , e , , . . . , e ,t , . . . , e k − , , . . . , e k − ,t ) is independent we set G = { e , , . . . , e ,t , e , , . . . , e ,t , . . . , e k − , , . . . , e k − ,t } . Assume to the contrary that the above tuple is not independent. Then there are two distinct T , T ∈F ( G ) such that σ ( T ) = σ ( T ). By symmetry, we may assume that T = 1 F ( G ) . There exist non-negative integers x , . . . , x t with x + . . . + x t = | T | such that T divides V x · . . . · V x t t in F ( G ), whence V x · . . . · V x t t T T − is a zero-sum sequence. Since V x · . . . · V x t t T T − has only one factorization and V , . . . , V t are the only atoms dividing V x · . . . · V x t t T T − , it follows that V x · . . . · V x t t T T − = V x · . . . · V x t t and hence T = T , a contradiction.Next we show that h g i , e i, , . . . , e i,t i ∩ h g j , e j, , . . . , e j,t : j ∈ [1 , k ] \ { i }i = { } for every i ∈ [1 , k ].Assume to the contrary that there exists 0 = h ∈ h g i , e i, , . . . , e i,t i ∩ h g j , e j, , . . . , e j,t : j ∈ [1 , k ] \ { i }i .Since h g i , e i, , . . . , e i,t i = [ g i , e i, , . . . , e i,t ] and h g j , e j, , . . . , e j,t : j ∈ [1 , k ] \ { i }i = [ g j , e j, , . . . , e j,t : j ∈ [1 , k ] \ { i } ], there exist a zero-sum free sequence T over { g i , e i, , . . . , e i,t } and a zero-sum free sequence T over { g j , e j, , . . . , e j,t : j ∈ [1 , k ] \ { i }} such that h = σ ( T ) = σ ( T ). Let N be large enough such that T divides U Ni . Then U Ni T T − is a zero-sum sequence such that supp( U Ni T T − ) ∩ { g j , e j, , . . . , e j,t : j ∈ [1 , k ] \ { i }} 6 = ∅ , which implies that there exists ν ∈ [1 , k ] \ { i } such that U ν divides U Ni T T − and hence U ν divides T , a contradiction. Therefore, we obtain that C ( H ∗ ) = h G P ∗ i = h e , , . . . , e ,t , g i ⊕ . . . ⊕ h e k, , . . . , e k,t , g k i . Let i ∈ [1 , k ] and set G i = h g i , e i, , . . . , e i,t i . Then G i ∼ = Z t ⊕ Z /m Z , where m ∈ N is the maximal orderof all the torsion elements of G i . Let gcd( s , s , . . . , s t ) = n . Then the fact that h = σ ( g s /ni e s /ni, · . . . · e s t /ni,t )has order n implies that n ≤ m . It remains to verify that n ≥ m . Let α ∈ G i such that ord( α ) = m .Suppose α = w g i + w e i, + . . . + w t e i,t , where w , . . . , w t ∈ N . Then ( g w i e w i, · . . . · e w t i,t ) m = U wi forsome w ∈ N with gcd( m, w ) = 1, which implies that m divides gcd( s , s , . . . , s t ) = n . Step 2.
Suppose that H ∗ ∼ = B ( G P ∗ ) and that G P ∗ has the given form. We have to show that B ( G P ∗ )is length-factorial but not factorial. We use the simple fact that if an abelian group G is a direct sum, say G = G ⊕ G , and if G ′ i ⊂ G i are subsets for i ∈ [1 , A ( G ′ ⊎ G ′ ) = A ( G ′ ) ⊎ A ( G ′ ). We define, for every i ∈ [0 , k ] and every j ∈ [1 , t ], U = g s e s , · . . . · e s t ,t , U i = e s i, · . . . · e s t i,t , V = g · . . . · g k , and V j = e ,j · . . . · e k,j . Clearly, we obtain that(2.3) A ( G P ∗ ) = { U , . . . , U k , V , . . . , V t } and U · . . . · U k = V s · . . . · V s t t . Thus, B ( G P ∗ ) is not factorial. By definition, we have | U i | = P tj =0 s j , | V j | = k + 1 for every i ∈ [0 , k ]and every j ∈ [0 , t ]. Assume to the contrary that there exists B ∈ B ( G P ∗ ) such that B has two distinctfactorizations of the same length. We may assume that B is a counterexample with minimal length.Suppose B = Y i ∈ I U a i i Y j ∈ J V b j j and B = Y i ∈ I U a ′ i i Y j ∈ J V b ′ j j are two distinct factorizations of the same length, where I , I ⊂ [0 , k ], J , J ⊂ [0 , t ], a i ∈ N forevery i ∈ I , a ′ i ∈ N for every i ∈ I , b j ∈ N for every j ∈ J , and b ′ j ∈ N for every j ∈ J .The minimality of | B | implies that I ∩ I = ∅ and J ∩ J = ∅ . If I ∪ I = ∅ , then those twofactorizations of B must be equal, a contradiction. By symmetry, we may suppose I = ∅ . Then ∪ i ∈ I supp( U i ) ⊂ supp( Q i ∈ I U a ′ i i Q j ∈ J V b ′ j j ) implies that J = [0 , t ] and J = ∅ , whence I = [0 , k ] and I = ∅ . It follows that B = k Y i =0 U a i i = t Y j =0 V b ′ j j , whence ( s + . . . + s t ) P ki =0 a i = P ki =0 a i | U i | = | B | = P tj =1 b ′ j | V j | = ( k + 1) P tj =1 b ′ j . Since s + . . . + s t = k + 1, we obtain P ki =0 a i = P tj =1 b ′ j , a contradiction to the fact that the two factorizations have the samelength. (cid:3) The system of sets of lengths L ( H ) of an atomic monoid H is said to be additively closed if the sumset L + L ∈ L ( H ) for all L , L ∈ L ( H ). Clearly, L + L = L implies that L = { } for all nonemptysets L , L ⊂ N , whence set addition is a unit-cancellative operation. Thus, L ( H ) is additively closed ifand only if ( L ( H ) , +) is a reduced monoid with set addition as operation.Let H be a Krull monoid class group G and let G ⊂ G denote the set of classes containing primedivisors. Then the inclusion B ( G ) ֒ → F ( G ) is a divisor homomorphism but it need not be a divisortheory ([51]). In Corollary 1.2 we prove that in case of length-factorial Krull monoids this inclusion is adivisor theory. Proof of Corollary 1.2.
Let H be a length-factorial Krull monoid, that is not factorial, and let all notationbe as in Theorem 1.1.1. Since the inclusion H ∗ ֒ → F ( P ∗ ) is a divisor theory, H ∗ ∼ = B ( G P ∗ ), and every class of G P ∗ contains precisely one prime divisor, the inclusion B ( G P ∗ ) ֒ → F ( G P ∗ ) is a divisor theory with class groupisomorphic to C ( H ∗ ) ∼ = C ( H ). The assertion on A ( G P ∗ ) follows from Equation (2.3).2. Let B ∈ B ( G P ∗ ) and z ∈ Z ( B ). By (2.3), z can be written in the form z = k Y i =0 U c i i t Y j =0 V d j j ∈ Z ( B ) , where c i , d j ∈ N for every i ∈ [0 , k ] and every j ∈ [0 , t ], and we have to determine the relations betweenthe exponents c , . . . , c k , d , . . . , d t . Let x = min { c i : i ∈ [0 , k ] } and x = min (cid:26)(cid:22) d j s j (cid:23) : j ∈ [0 , t ] (cid:27) . CHARACTERIZATION OF LENGTH-FACTORIAL KRULL MONOIDS 13
Then z = k Y i =0 U c i i t Y j =0 V d j j = ( U · . . . · U k ) x ( V s · . . . · V s t t ) x k Y i =0 U c i − x i t Y j =0 V d j − x s j j . We set x = x + x , y i = c i − x , and z j = d j − x s j for every i ∈ [0 , k ] and every j ∈ [0 , t ]. Thus, B = ( U · . . . · U k ) x k Y i =0 U y i i t Y j =0 V z j j has a factorization of the required form. Since for every ν ∈ [0 , x ], z ′ = ( U · . . . · U k ) ν ( V s · . . . · V s t t ) x − ν k Y i =0 U y i i t Y j =0 V z j j ∈ Z ( B ) , we have | z | ∈ k X i =0 y i + t X j =0 z j + n ν ( k + 1) + ( x − ν ) t X j =0 s j : ν ∈ [0 , x ] o ⊂ L ( B ) . If B can be written uniquely in the asserted form then, since z is chosen arbitrary, it follows that L ( B ) = k X i =0 y i + t X j =0 z j + n ν ( k + 1) + ( x − ν ) t X j =0 s j : ν ∈ [0 , x ] o . It remains to verify the uniqueness assertion. Suppose that B = ( U · . . . · U k ) x k Y i =0 U y i i t Y j =0 V z j j = ( U · . . . · U k ) x ′ k Y i =0 U y ′ i i t Y j =0 V z ′ j j , where • x, y , . . . , y k , z , . . . , z t ∈ N , y i = 0 for some i ∈ [0 , k ], and z j < s j for some j ∈ [0 , t ], and • x ′ , y ′ , . . . , y ′ k , z ′ , . . . , z ′ t ∈ N , y ′ i = 0 for some i ∈ [0 , k ], and z ′ j < s j for some j ∈ [0 , t ].Note, if there would exist i ∈ [0 , k ] such that U i divides Q tj =1 V z ′ j j , then s j = v e i,j ( U i ) ≤ v e i,j ( t Y j =1 V z ′ j j ) = z j ′ , a contradiction. If x > x ′ , then U y i +1 i divides Q ki =1 U y ′ i i Q tj =1 V z ′ j j . Since supp( U i ) ∩ supp( U i ) = ∅ for every i ∈ [0 , k ] \ { i } , we have U y i +1 i divides U y ′ i i Q tj =1 V z ′ j j = Q tj =1 V z ′ j j , a contradiction. Thus x ≤ x ′ . By symmetry, we obtain that x ′ ≤ x , whence x = x ′ . If y i > y ′ i for some i ∈ [0 , k ], then U i must divide Q tj =1 V z ′ j j , a contradiction. Thus y i ≤ y ′ i for every i ∈ [0 , k ]. By symmetry, we obtain that y ′ i ≤ y i , whence y i = y ′ i for every i ∈ [0 , k ]. Since x = x ′ and y i = y ′ i for every i ∈ [0 , k ], we infer that Q tj =1 V z j j = Q tj =1 V z ′ j j , whence z j = z ′ j for every j ∈ [0 , t ].3. By Lemma 2.3.3, Lemma 2.4.1, and Theorem 1.1, we have L ( H ) = L (cid:0) B ( G P ∗ ) (cid:1) . By item 2., weinfer that L ( H ) ⊂ n(cid:8) y + ν ( k + 1) + ( x − ν ) t X j =0 s j : ν ∈ [0 , x ] (cid:9) : y, x ∈ N o . Conversely, if y, x ∈ N and B = ( U · . . . · U k ) x U y , then { y + ν ( k + 1) + ( x − ν ) P tj =0 s j : ν ∈ [0 , x ] } = L ( B ) ∈ L ( H ), whence L ( H ) = n(cid:8) y + ν ( k + 1) + ( x − ν ) t X j =0 s j : ν ∈ [0 , x ] (cid:9) : y, x ∈ N o . The given description shows immediately that L ( H ) is additively closed with respect to set addition. (cid:3) Before proving Corollary 1.3 we briefly recall the involved concepts. Let H be a Krull monoid and H red ֒ → F = F ( P ) be a divisor theory. Then H satisfies the approximation property if one of the followingequivalent conditions is satisfied ([28, Proposition 2.5.2]):(a) For all n ∈ N and distinct p, p , . . . , p n ∈ P there exists some a ∈ H such that v p ( a ) = 1 and v p i ( a ) = 0 for all i ∈ [1 , n ].(b) For all a, b ∈ F , there exists some c ∈ F such that [ a ] = [ c ] ∈ G and gcd( b, c ) = 1. Proof of Corollary 1.3.
Let H be a Krull monoid. Without restriction we may suppose that H is reduced.Using the notation of Theorem 1.1, we have H = F ( P ) × H ∗ and a divisor theory F ( P ) × H ∗ ֒ →F ( P ) × F ( P ∗ ). Let G P ∗ ⊂ C ( H ∗ ) ∼ = C ( H ) denote the set of classes containing prime divisors.1. If H and H ∗ are length-factorial but not factorial, then P ∗ is finite by Theorem 1.1. Thus, Condition(b) above cannot hold, whence H does not satisfy the approximation property.2. Suppose that every nonzero class of G = C ( H ∗ ) contains a prime divisor. Note that 0 ∈ A ( G ) isthe only prime element of B ( G ) and B ( G ) = F ( { } ) × B ( G \ { } ). Thus B ( G ) is length-factorial if andonly if B ( G \ { } ) is length-factorial.First, we suppose that H ∗ ∼ = B ( G \ { } ) and that either | G | ≤ G ∼ = C ⊕ C . We have to verifythat B ( G ) is length-factorial. If | G | ≤
2, then B ( G ) is factorial and hence length-factorial. If | G | = 3 or G ∼ = C ⊕ C , then it can be checked directly that B ( G ) is length-factorial.Conversely, suppose that H ∗ is length-factorial. Since G \ { } ⊂ G P ∗ , the description of G P ∗ achievedin Theorem 1.1 implies that | G | ≤ G ∼ = C ⊕ C . (cid:3) In order to prove Corollary 1.4, we first gather some basics from the theory of rings with zero-divisors.Let R be a commutative ring with identity and let R • denote its monoid of regular elements. Then R is additively regular if for each pair of elements a, b ∈ R with b regular, there is an element r ∈ R such that a + br is a regular element of R ([42, 43]). Every additively regular ring is a Marot ring and every Marotring is a v -Marot ring. The ring R is a Krull ring if it is completely integrally closed and satisfies theACC on regular divisorial ideals. If R is a Krull ring, then R • is a Krull monoid and if R is a v -Marotring, then the converse holds ([29, Theorem 3.5]). We say that R is atomic (factorial, half-factorial, resp.length-factorial) if R • has the respective property.Next we need the concept of normalizing Krull rings. A cancellative but not necessarily commutativesemigroup S (resp. a ring R ) is said to be normalizing if aS = Sa for all a ∈ S (resp. aR = Ra for all a ∈ R ). A prime Goldie ring is said to be a Krull ring (or a Krull order) if it is completelyintegrally closed (equivalently, a maximal order) and satisfies the ACC on two-sided divisorial ideals.Thus, every commutative Krull domain is a normalizing Krull ring. For examples and background onnon-commutative (normalizing) Krull rings we refer to [53, 10, 45, 1], and for background on factorizationsin the non-commutative setting to [4, 52]. In particular, normalizing Krull monoids are transfer Krull. Proof of Corollary 1.4.
1. Let R be an additively regular Krull ring. Then R • satisfies the approximationproperty by [46, Theorem 2.2] (this needs the assumption that R is additively regular). Thus R • is aKrull monoid satisfying the approximation property, whence the assertion follows from Corollary 1.3.1.2. Let R be a normalizing Krull ring. Then R satisfies the approximation property ([10, Proposition2.9], [44, Theorem 4]). If H denotes the monoid of regular elements, then H red is a commutative Krullmonoid by [26, Corollary 4.14 and Proposition 5.1]. Thus, the assertion follows from Corollary 1.3.1. (cid:3) Proof of Corollary 1.5.
Let H be a length-factorial transfer Krull monoid. We have to show that H red is a Krull monoid. Since H red is a length-factorial transfer Krull monoid, we may suppose that H isreduced. Let B be a Krull monoid and let θ ′ : H → B be a transfer homomorphism. We may supposethat B is reduced and start with the following assertion. A. H is cancellative. CHARACTERIZATION OF LENGTH-FACTORIAL KRULL MONOIDS 15
Proof of A . Let a, b, c ∈ H such that ab = ac . Since θ ′ ( a ) θ ′ ( b ) = θ ′ ( a ) θ ′ ( c ), we obtain that θ ′ ( b ) = θ ′ ( c ). If θ ′ ( b ) = θ ′ ( c ) = 1 B , then b = c = 1 H . If θ ′ ( b ) = θ ′ ( c ) = w · . . . · w r , where r ∈ N and w , . . . , w r ∈ A ( B ),then there exist b , . . . , b r , c , . . . , c r ∈ A ( H ) such that b = b · . . . · b r and c = c · . . . · c r . Suppose a = a · . . . · a k , where k ∈ N and a , . . . , a k ∈ A ( H ). Then the two factorizations z = a · . . . · a k b · . . . · b r ∈ Z ( ab ) and z = a · . . . · a k c · . . . · c r ∈ Z ( ab ) of ab have the same length k + r , whence z = z . Thus b · . . . · b r = c · . . . · c r ∈ Z ( H ), whence b = c ∈ H . (cid:3) (Proof of A ). Thus, H is a reduced cancellative length-factorial transfer Krull monoid. If H is factorial, then H isKrull. Suppose that H is not factorial. Then H is not half-factorial. Let G be the class group of B and let G ⊂ G be the set of classes containing prime divisors. Since H is not factorial, it is not half-factorial. Thus, Lemma 2.3 implies that B is length-factorial but not half-factorial. Theorem 1.1 impliesthat every class of G contains precisely one prime divisor. Lemma 2.4.1 implies that there is a transferhomomorphism β : B → B ( G ). Since every class of G contains precisely one prime divisor, Lemma2.4.3 implies that G = [ G \ { g } ] for every g ∈ G . Since the composition of transfer homomorphisms isa transfer homomorphism again, we obtain a transfer homomorphism θ = β ◦ θ ′ : H → B ( G ).Let P ⊂ H be the set of prime elements of H and H = { a ∈ H : p ∤ a for all p ∈ P } . Since H iscancellative, we obtain that H = F ( P ) × H . Since G = [ G \ { g } ] for every g ∈ G , the only possibleprime element of B ( G ) is the sequence S = 0 ∈ F ( G ). Thus Lemma 2.3.2 implies that, if P = ∅ ,then θ ( P ) = { } . Thus, we obtain that θ ( H ) = B ( G \ { } ) and hence θ H : H → B ( G \ { } ) is asurjective transfer homomorphism. By Lemma 2.3, B ( G \ { } ) is length-factorial but not half-factorial.By Corollary 1.2.1, A ( G \{ } ) is finite, say A ( G \{ } ) = { U ′ , . . . , U ′ k , V ′ , . . . , V ′ ℓ } , U ′ · . . . · U ′ k = V ′ · . . . · V ′ ℓ , k, ℓ ∈ N ≥ , k = ℓ , and U ′ i = V ′ j for all i ∈ [1 , k ] and j ∈ [1 , ℓ ].Assume to the contrary that θ H is not injective. Then there exist a, b ∈ H with a = b such that T = θ ( a ) = θ ( b ), say T = W · . . . · W r , where r ∈ N and W , . . . , W r ∈ A ( G \ { } ). Then there exist a , . . . , a r , b , . . . , b r ∈ A ( H ) such that a = a · . . . · a r , b = b · . . . · b r , and θ ( a i ) = θ ( b i ) = W i forall i ∈ [1 , r ]. Since a = b , there exists i ∈ [1 , r ], say i = 1, such that a = b . After renumbering ifnecessary, we may suppose W = U ′ . Let c ∈ H such that θ ( c ) = Q ki =2 U ′ i . Therefore, θ ( a c ) = θ ( b c ) = U ′ · . . . · U ′ k = V ′ · . . . · V ′ ℓ , which implies that there exist u , . . . , u ℓ , v , . . . , v ℓ such that a c = u · . . . · u ℓ , b c = v · . . . · v ℓ , and θ ( u j ) = θ ( v j ) = V ′ j for all j ∈ [1 , ℓ ]. We observe that a b c = a v · . . . · v ℓ = b u · . . . · u ℓ . If there exists j ∈ [1 , ℓ ] such that a = u j , then U ′ = θ ( a ) = θ ( u j ) = V ′ j , a contradiction. Thus a b c has two distinct factorization of length ℓ + 1, a contradiction. Therefore θ H is injective, whence H ∼ = B ( G \ { } ) is Krull and so H = F ( P ) × H is Krull. (cid:3) The monoids, discussed in Example 2.2.2, are reduced and length-factorial but not cancellative. Thusthey cannot be transfer Krull by Corollary 1.5. We end with an example of transfer Krull monoids.
Example 2.7.
Let R be a Bass ring and let T ( R ) be the monoid of isomorphism classes of torsion-freefinitely generated R -modules, together with the operation induced by the direct sum (this is a monoidas discussed in Example 2.2.1). Then T ( R ) is a reduced transfer Krull monoid by [3, Theorem 1.1].There are algebraic characterizations of when T ( R ) is factorial, resp. half-factorial, resp. cancellative(see [3, Proposition 3.13, Corollary 1.2, Remark 3.17]). These characterizations show that T ( R ) is rarelycancellative, whence rarely Krull, and thus, by Corollary 1.3, it is rarely length-factorial. References [1] E. Akalan and H. Marubayashi,
Multiplicative Ideal Theory in Non-Commutative Rings , in Multiplicative Ideal Theoryand Factorization Theory, Springer, 2016, pp. 1 – 21.[2] N.R. Baeth and A. Geroldinger,
Monoids of modules and arithmetic of direct-sum decompositions , Pacific J. Math. (2014), 257 – 319.[3] N.R. Baeth and D. Smertnig,
Lattices over Bass rings and graph agglomerations , Algebras and Representation Theory,to appear, https://arxiv.org/abs/2006.10002.[4] ,
Factorization theory: From commutative to noncommutative settings , J. Algebra (2015), 475 – 551.[5] N.R. Baeth and R. Wiegand,
Factorization theory and decomposition of modules , Am. Math. Mon. (2013), 3 – 34.[6] E.R. Garc´ıa Barroso, I. Garc´ıa-Marco, and I. M´arquez-Corbella,
Factorizations of the same lengths in abelian monoids ,Ricerche di Matematica, to appear, https://arxiv.org/abs/2007.05567.[7] G. M. Bergman,
Coproducts and some universal ring constructions , Trans. Amer. Math. Soc. (1974), 33–88.[8] G. M. Bergman and W. Dicks,
Universal derivations and universal ring constructions , Pacific J. Math. (1978),no. 2, 293–337.[9] W. Bruns and J. Gubeladze, Polytopes, Rings, and K-Theory , Springer, 2009.[10] L. Le Bruyn and F. van Oystaeyen,
A note on noncommutative Krull domains , Commun. Algebra (1986), 1457 –1472.[11] S.T. Chapman, J. Coykendall, F. Gotti, and W.W. Smith, Length-factoriality in commutative monoids and integraldomains , J. Algebra, to appear, https://arxiv.org/abs/2101.05441.[12] S.T. Chapman, P.A. Garc´ıa-S´anchez, D. Llena, and J. Marshall,
Elements in a numerical semigroup with factorizationsof the same length , Canad. Math. Bull. (2011), 39 – 43.[13] S.T. Chapman, U. Krause, and E. Oeljeklaus, On Diophantine monoids and their class groups , Pacific J. Math. (2002), 125 – 147.[14] J. Correa-Morris and F. Gotti,
On the additive structure of algebraic valuations of cyclic free semirings ,https://arxiv.org/abs/2008.13073.[15] J. Coykendall and W.W. Smith,
On unique factorization domains , J. Algebra (2011), 62 – 70.[16] A. Facchini,
Direct sum decomposition of modules, semilocal endomorphism rings, and Krull monoids , J. Algebra (2002), 280 – 307.[17] ,
Krull monoids and their application in module theory , Algebras, Rings and their Representations (A. Facchini,K. Fuller, C. M. Ringel, and C. Santa-Clara, eds.), World Scientific, 2006, pp. 53 – 71.[18] ,
Direct-sum decompositions of modules with semilocal endomorphism rings , Bull. Math. Sci. (2012), 225 –279.[19] , Semilocal categories and modules with semilocal endomorphism rings , Progress in Mathematics, vol. 331,Birkh¨auser/Springer, Cham, 2019.[20] A. Facchini, W. Hassler, L. Klingler, and R. Wiegand,
Direct-sum decompositions over one-dimensional Cohen-Macaulay local rings , in Multiplicative Ideal Theory in Commutative Algebra, Springer, 2006, pp. 153 – 168.[21] A. Facchini and R. Wiegand,
Direct-sum decomposition of modules with semilocal endomorphism rings , J. Algebra (2004), 689 – 707.[22] V. Fadinger and D. Windisch,
On the distribution of prime divisors in Krull monoid algebras ,https://arxiv.org/abs/2101.04398.[23] Y. Fan, A. Geroldinger, F. Kainrath, and S. Tringali,
Arithmetic of commutative semigroups with a focus on semigroupsof ideals and modules , J. Algebra Appl. (2017), 1750234 (42 pages).[24] A. Foroutan, Monotone chains of factorizations , Focus on commutative rings research (A. Badawi, ed.), Nova Sci.Publ., New York, 2006, pp. 107 – 130.[25] A. Garc´ıa-Elsener, P. Lampe, and D. Smertnig,
Factoriality and class groups of cluster algebras , Advances in Math. (2019), 106858, 48.[26] A. Geroldinger,
Non-commutative Krull monoids: a divisor theoretic approach and their arithmetic , Osaka J. Math. (2013), 503 – 539.[27] A. Geroldinger, D.J. Grynkiewicz, G.J. Schaeffer, and W.A. Schmid, On the arithmetic of Krull monoids with infinitecyclic class group , J. Pure Appl. Algebra (2010), 2219 – 2250.[28] A. Geroldinger and F. Halter-Koch,
Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory , Pureand Applied Mathematics, vol. 278, Chapman & Hall/CRC, 2006.[29] A. Geroldinger, S. Ramacher, and A. Reinhart, On v -Marot Mori rings and C -rings , J. Korean Math. Soc. (2015),1 – 21.[30] A. Geroldinger and A. Reinhart, The monotone catenary degree of monoids of ideals , Internat. J. Algebra Comput. (2019), 419 – 457.[31] A. Geroldinger and W.A. Schmid, A characterization of Krull monoids for which sets of lengths are (almost) arith-metical progressions , Rev. Mat. Iberoam. (2021), 293 – 316. CHARACTERIZATION OF LENGTH-FACTORIAL KRULL MONOIDS 17 [32] A. Geroldinger, W.A. Schmid, and Q. Zhong,
Systems of sets of lengths: transfer Krull monoids versus weakly Krullmonoids , in Rings, Polynomials, and Modules, Springer, Cham, 2017, pp. 191 – 235.[33] A. Geroldinger and P. Yuan,
The monotone catenary degree of Krull monoids , Result. Math. (2013), 999 – 1031.[34] A. Geroldinger and Q. Zhong, Factorization theory in commutative monoids , Semigroup Forum (2020), 22 – 51.[35] R. Gilmer,
Some questions for further research , in Multiplicative Ideal Theory in Commutative Algebra, Springer,2006, pp. 405 – 415.[36] K. R. Goodearl, von Neumann regular rings , Monographs and Studies in Mathematics, vol. 4, Pitman (AdvancedPublishing Program), Boston, Mass.-London, 1979.[37] F. Gotti,
Geometric and combinatorial aspects of submonoids of a finite-rank free commutative monoid , Linear AlgebraAppl. (2020), 146–186.[38] ,
Irreducibility and factorizations in monoid rings , In: Numerical Semigroups, vol. 40, Springer INdAM Series,2020, pp. 129 – 139.[39] D.J. Grynkiewicz,
The characterization of finite elasticities , http://diambri.org/FiniteRhoChar-0076.pdf.[40] W. Hassler,
Properties of factorizations with successive lengths in one-dimensional local domains , J. Commut. Algebra (2009), 237 – 268.[41] W. Hassler, R. Karr, L. Klingler, and R. Wiegand, Large indecomposable modules over local rings , J. Algebra (2006), 202 – 215.[42] J.A. Huckaba,
Commutative rings with zero divisors , Pure and Applied Mathematics, vol. 117, Marcel Dekker, 1988.[43] T.G. Lucas,
Additively regular rings and Marot rings , Palestine J. Math. (2016), 90 – 99.[44] G. Maury, Th´eor`eme d’approximation pour un anneau de Krull non commutatif (au sens de M. Chamarie) et appli-cations , Arch. Math. (Basel) (1982), no. 6, 541–545.[45] J. Okni´nski, Noetherian Semigroup Algebras and Beyond , in Multiplicative Ideal Theory and Factorization Theory,Springer, 2016, pp. 255 – 276.[46] E. Osmanagi´c,
On an approximation theorem for Krull rings with zero divisors , Commun. Algebra (1999), 3647 –3657.[47] A. Philipp, A characterization of arithmetical invariants by the monoid of relations II: The monotone catenary degreeand applications to semigroup rings , Semigroup Forum (2015), 220 – 250.[48] A. Plagne and W.A. Schmid, On large half-factorial sets in elementary p -groups : maximal cardinality and structuralcharacterization , Isr. J. Math. (2005), 285 – 310.[49] , On the maximal cardinality of half-factorial sets in cyclic groups , Math. Ann. (2005), 759 – 785.[50] ,
On congruence half-factorial Krull monoids with cyclic class group , Journal of Combinatorial Algebra (2020), 331 – 400.[51] W.A. Schmid, Higher-order class groups and block monoids of Krull monoids with torsion class group , J. AlgebraAppl. (2010), 433 – 464.[52] D. Smertnig, Factorizations of elements in noncommutative rings: A Survey , in Multiplicative Ideal Theory andFactorization Theory, Springer, 2016, pp. 353 – 402.[53] P. Wauters and E. Jespers,
Examples of noncommutative Krull rings , Commun. Algebra (1986), 819 – 832.[54] Q. Zhong, Sets of minimal distances and characterizations of class groups of Krull monoids , Ramanujan J. (2018),719 – 737. University of Graz, NAWI Graz, Institute for Mathematics and Scientific Computing, Heinrichstraße 36,8010 Graz, AustriaSchool of Mathematics and statistics, Shandong University of Technology, Zibo, Shandong 255000, China
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