Direct sum behavior of lattices over sigma-I rings
type: Item
instance of: scholarly article
title: Direct sum behavior of lattices over sigma-I rings
zbMATH Open document ID: 0656.13016
DOI: 10.1016/0022-4049(88)90084-9
author: Jeremy Haefner
published in: Journal of Pure and Applied Algebra
publication date: 1988-01-01T00:00:00Z
review text: A ring-order is a commutative Noetherian reduced ring R of Krull dimension one whose integral closure is a finitely generated R-module. A \(\Sigma\) I-ring is a ring-order in which every lattice (i.e. finitely generated submodule of a free R-module) is isomorphic to a direct sum of ideals. For example, a Bass ring (i.e. a ring-order in which every ideal is doubly generated) is a \(\Sigma\) I-ring. This paper considers what uniqueness is inherent in a lattice's direct sum decomposition into undecomposable lattices. It is shown that a \(\Sigma\) I-ring has the property that every lattice has a unique decomposition into a direct sum of indecomposable lattices if and only if R has at most one singular maximal ideal and Pic(R) is trivial. To handle the global case, the genus class group G(M) of a lattice M is studied. It is shown that G(M) is a homomorphic image of a direct sum of t copies of Pic(R) where t depends only on R. The \(\Sigma\) I-rings in which every lattice has a unique number of indecomposables summands are characterized.
Mathematics Subject Classification ID: 13E15
Mathematics Subject Classification ID: 13C05
zbMATH DE Number: 4071060
zbMATH Keywords: genus class group
zbMATH Keywords: ring-order
zbMATH Keywords: \(\Sigma \) I-ring
zbMATH Keywords: lattice
zbMATH Keywords: Bass ring
reviewed by: D. D. Anderson
MaRDI profile type: Publication
full work available at URL: 0022-4049(88)90084-9
OpenAlex ID: W1980983339
cites work: On the ubiquity of Gorenstein rings
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cites work: Q3797295
description: scientific article; zbMATH DE number 4071060