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The Monotone Catenary Degree of Krull Monoids
, 2013
"... Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. The monotone catenary degree cmon(H) of H is the smallest integer m with the following property: for each a ∈ H and each two factorizations z, z ′ of a with length z  ≤ z′, there exist factorizati ..."
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Cited by 15 (9 self)
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Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. The monotone catenary degree cmon(H) of H is the smallest integer m with the following property: for each a ∈ H and each two factorizations z, z ′ of a with length z  ≤ z′, there exist factorizations z = z0,..., zk = z ′ of a with increasing lengths—that is, z0  ≤ · · · ≤ zk—such that, for each i ∈ [1, k], zi arises from zi−1 by replacing at most m atoms from zi−1 by at most m new atoms. Up to now there was only an abstract finiteness result for cmon(H), but the present paper offers the first explicit upper and lower bounds for cmon(H) in terms of the group invariants of G.
Monoids of modules and arithmetic of directsum decompositions
 Pacific J. Math
"... Abstract. Let R be a (possibly noncommutative) ring and let C be a class of finitely generated (right) Rmodules which is closed under finite direct sums, direct summands, and isomorphisms. Then the set V(C) of isomorphism classes of modules is a commutative semigroup with operation induced by the d ..."
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Cited by 10 (7 self)
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Abstract. Let R be a (possibly noncommutative) ring and let C be a class of finitely generated (right) Rmodules which is closed under finite direct sums, direct summands, and isomorphisms. Then the set V(C) of isomorphism classes of modules is a commutative semigroup with operation induced by the direct sum. This semigroup encodes all possible information about direct sum decompositions of modules in C. If the endomorphism ring of each module in C is semilocal, then V(C) is a Krull monoid. Although this fact was observed nearly a decade ago, the focus of study thus far has been on ring and moduletheoretic conditions enforcing that V(C) is Krull. If V(C) is Krull, its arithmetic depends only on the class group of V(C) and the set of classes containing prime divisors. In this paper we provide the first systematic treatment to study the directsum decompositions of modules using methods from Factorization Theory of Krull monoids. We do this when C is the class of finitely generated torsionfree modules over certain one and twodimensional commutative Noetherian local rings. 1.
Arithmetic of seminormal weakly Krull monoids and domains
 J. Algebra
"... Abstract. We study the arithmetic of seminormal vnoetherian weakly Krull monoids with nontrivial conductor which have finite class group and prime divisors in all classes. These monoids include seminormal orders in holomorphy rings in global fields. The crucial property of seminormality allows us ..."
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Abstract. We study the arithmetic of seminormal vnoetherian weakly Krull monoids with nontrivial conductor which have finite class group and prime divisors in all classes. These monoids include seminormal orders in holomorphy rings in global fields. The crucial property of seminormality allows us to give precise arithmetical results analogous to the wellknown results for Krull monoids having finite class group and prime divisors in each class. This allows us to show, for example, that unions of sets of lengths are intervals and to provide a characterization of halffactoriality. 1.