scholarly journals The system of sets of lengths in Krull monoids under set addition

2016 ◽  
Vol 32 (2) ◽  
pp. 571-588 ◽  
Author(s):  
Alfred Geroldinger ◽  
Wolfgang Schmid
Keyword(s):  
Krull Monoids ◽  
Sets Of Lengths ◽  
Set Addition

scholarly journals Factorization theory in commutative monoids

2020 ◽  
Vol 100 (1) ◽  
pp. 22-51 ◽  
Author(s):  
Alfred Geroldinger ◽  
Qinghai Zhong
Keyword(s):  
Algebraic Number ◽  
Number Fields ◽  
Finitely Generated ◽  
Commutative Monoids ◽  
Algebraic Number Fields ◽  
Factorization Theory ◽  
Krull Monoids ◽  
Sets Of Lengths ◽  
Set Addition ◽  
Finiteness Properties

AbstractThis is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and their catenary degrees. We also provide examples where these finiteness properties do not hold.

scholarly journals Long Sets of Lengths With Maximal Elasticity

2018 ◽  
Vol 70 (6) ◽  
pp. 1284-1318 ◽  
Author(s):  
Alfred Geroldinger ◽  
Qinghai Zhong
Keyword(s):  
Finite Type ◽  
Algebraic Number ◽  
Class Group ◽  
Number Fields ◽  
Additive Combinatorics ◽  
Algebraic Number Fields ◽  
Krull Monoids ◽  
Sets Of Lengths ◽  
Positive Integers ◽  
Krull Domains

AbstractWe introduce a newinvariant describing the structure of sets of lengths in atomicmonoids and domains. For an atomic monoid H, let Δρ(H) be the set of all positive integers d that occur as differences of arbitrarily long arithmetical progressions contained in sets of lengths havingmaximal elasticity ρ(H). We study Δρ(H) for transfer Krull monoids of finite type (including commutative Krull domains with finite class group) with methods from additive combinatorics, and also for a class of weakly Krull domains (including orders in algebraic number fields) for which we use ideal theoretic methods.

scholarly journals Systems of Sets of Lengths: Transfer Krull Monoids Versus Weakly Krull Monoids

2017 ◽  
pp. 191-235 ◽  
Author(s):  
Alfred Geroldinger ◽  
Wolfgang A. Schmid ◽  
Qinghai Zhong
Keyword(s):  
Krull Monoids ◽  
Sets Of Lengths

scholarly journals Factorizations in Bounded Hereditary Noetherian Prime Rings

2018 ◽  
Vol 62 (2) ◽  
pp. 395-442 ◽  
Author(s):  
Daniel Smertnig
Keyword(s):  
Class Group ◽  
Structure Theory ◽  
Projective Modules ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Prime Rings ◽  
Sets Of Lengths ◽  
Irreducible Elements ◽  
Transfer Homomorphism ◽  
Zero Sum

AbstractIf H is a monoid and a = u1 ··· uk ∈ H with atoms (irreducible elements) u1, … , uk, then k is a length of a, the set of lengths of a is denoted by Ⅼ(a), and ℒ(H) = {Ⅼ(a) | a ∈ H} is the system of sets of lengths of H. Let R be a hereditary Noetherian prime (HNP) ring. Then every element of the monoid of non-zero-divisors R• can be written as a product of atoms. We show that if R is bounded and every stably free right R-ideal is free, then there exists a transfer homomorphism from R• to the monoid B of zero-sum sequences over a subset Gmax(R) of the ideal class group G(R). This implies that the systems of sets of lengths, together with further arithmetical invariants, of the monoids R• and B coincide. It is well known that commutative Dedekind domains allow transfer homomorphisms to monoids of zero-sum sequences, and the arithmetic of the latter has been the object of much research. Our approach is based on the structure theory of finitely generated projective modules over HNP rings, as established in the recent monograph by Levy and Robson. We complement our results by giving an example of a non-bounded HNP ring in which every stably free right R-ideal is free but which does not allow a transfer homomorphism to a monoid of zero-sum sequences over any subset of its ideal class group.

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