The structure of sets of lengths

2006 ◽  
pp. 257-322
Keyword(s):  
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.

scholarly journals The monotone catenary degree of monoids of ideals

2019 ◽  
Vol 29 (03) ◽  
pp. 419-457 ◽  
Author(s):  
Alfred Geroldinger ◽  
Andreas Reinhart
Keyword(s):  
Ideal Theory ◽  
Integral Domains ◽  
Finiteness Conditions ◽  
Central Topic ◽  
Sets Of Lengths ◽  
Catenary Degree

Factoring ideals in integral domains is a central topic in multiplicative ideal theory. In the present paper, we study monoids of ideals and consider factorizations of ideals into multiplicatively irreducible ideals. The focus is on the monoid of nonzero divisorial ideals and on the monoid of [Formula: see text]-invertible divisorial ideals in weakly Krull Mori domains. Under suitable algebraic finiteness conditions, we establish arithmetical finiteness results, in particular, for the monotone catenary degree and for the structure of sets of lengths and of their unions.

scholarly journals A structure theorem for sets of lengths

1998 ◽  
Vol 78 (2) ◽  
pp. 225-259 ◽  
Author(s):  
Alfred Geroldinger
Keyword(s):  
Structure Theorem ◽  
Sets Of Lengths

scholarly journals Sets of lengths in atomic unit-cancellative finitely presented monoids

2018 ◽  
Vol 151 (2) ◽  
pp. 171-187 ◽  
Author(s):  
Alfred Geroldinger ◽  
Emil Daniel Schwab
Keyword(s):  
Atomic Unit ◽  
Sets Of Lengths ◽  
Finitely Presented

scholarly journals On the asymptotic behavior of unions of sets of lengths in atomic monoids

2008 ◽  
Vol 1 (1) ◽  
pp. 101-110 ◽  
Author(s):  
Paul Baginski ◽  
Scott Chapman ◽  
Natalie Hine ◽  
João Paixão
Keyword(s):  
Asymptotic Behavior ◽  
Sets Of Lengths
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