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Factorizations in Bounded Hereditary Noetherian Prime Rings

Part of: Grothendieck groups and $K_0$ Homological methods Conditions on elements Semigroups Chain conditions, growth conditions, and other forms of finiteness

Published online by Cambridge University Press:  19 November 2018

Daniel Smertnig
Daniel Smertnig*
Affiliation:
University of Graz, NAWI Graz, Institute for Mathematics and Scientific Computing, Heinrichstraße 36, 8010 Graz, Austria ([email protected])
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Abstract

If 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.

Keywords

Hereditary Noetherian prime ringsDedekind prime ringsKrull monoidsfactorization theorymonoids of zero-sum sequencessets of lengthscatenary degreestransfer homomorphismsideal class groups

MSC classification

Primary: 16E60: Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc.
Secondary: 16P40: Noetherian rings and modules 16U30: Divisibility, noncommutative UFDs 19A49: $K_0$ of other rings 20M13: Arithmetic theory of monoids
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 2 , May 2019 , pp. 395 - 442
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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