Primal, completely irreducible, and primary meet decompositions in modules
type: Item
instance of: scholarly article
zbMATH Open document ID: 1265.13010
publication date: 2012-10-04T00:00:00Z
Mathematics Subject Classification ID: 13C13
Mathematics Subject Classification ID: 13C99
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 16D80
zbMATH DE Number: 6090192
zbMATH Keywords: meet irreducible
zbMATH Keywords: primal module
zbMATH Keywords: completely irreducible
zbMATH Keywords: coirreducible
zbMATH Keywords: completely coirreducible
zbMATH Keywords: coprimary
zbMATH Keywords: \(p\)-primary submodule
zbMATH Keywords: rich in coirreducibles
author: Toma Albu
author: Patrick F. Smith
MaRDI profile type: Publication
title: Primal, completely irreducible, and primary meet decompositions in modules
published in: Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie. Nouvelle Série
review text: Inspired by work of \textit{L. Fuchs}, \textit{W. Heinzer} and \textit{B. Olberding} [Trans. Am. Math. Soc. 357, No. 7, 2771--2798 (2005; Zbl 1066.13003); ibid. 358, No. 7, 3113--3131 (2006; Zbl 1091.13011)] on primal and completely irreducible ideals , this paper studies similar and related concepts for modules. A proper submodule \(N\) of \(M\) is called (meet) irreducible if \(N=K\cap L\) for submodules \(K\) and \(L\) of \(M\), implies that \(N=K\) or \(N=L\). A proper submodule \(N\) of \(M\) is said to be completely irreducible (CI) if \(N\) is not the intersection of any collection of submodules of \(M\) each properly containing \(N\). A module \(M\) is called coirreducible (resp. completely coirreducible) if 0 is an irreducible (CI) submodule of \(M\). A proper submodule \(N\) of an \(R\)-module \(M\) over a commutative ring is said to be a primary submodule of \(M\) if \(rm\in N\) for \(r\in\), \(m\in M\), implies that \(m\in N\) or \(r\in R_m (N)\) (\(=\{r\in R|\exists k\in\mathbb{R}{\text{ such that }}r^k M\subseteq N\}\)). For a few instances, such as in the case where \(M\) is a module with primary decomposition, if \(M\) is a Laskerian module or if \(R\) has classical Krull dimension 0, it is shown that every CI submodule of \(M\) is \(\mathbf m\)-primary for some \({\mathbf m}\in \mathrm{Max}(R)\). In the case of a Noetherian ring, a submodule of \(M\) is CI if and only if it is an irreducible \(\mathbf p\)-primary submodule for some \({\mathbf p}\in \mathrm{Max}(R)\).NEWLINENEWLINE A submodule \(N\) of \(M\) over a commutative ring \(R\) is said to be primal if \(N\neq M\) and the set \(Z(M/N)\) of all zero divisors of \(M/N\) is an ideal of \(R\); \(M\) is called coprimal if \(0\) is a primal submodule of \(M\). For a Noetherian ring \(R\), several equivalences are found for every primal submodule (resp. ideal) of \(R\) to be primary.NEWLINENEWLINE Let \(A\leq C\) be submodules of \(M_R\). Then \(A\) is a relevant completely irreducible divisor (RCI…
reviewed by: Frieda Theron
description: scientific article; zbMATH DE number 6090192