A characterization of prime submodules
type: Item
instance of: scholarly article
title: A characterization of prime submodules
zbMATH Open document ID: 0924.13003
published in: Journal of Algebra
publication date: 1999-05-26T00:00:00Z
full work available at URL: http://hdl.handle.net/11655/18917
review text: Let \(R\) be a commutative ring (with non-zero identity) and let \(M\) be an \(R\)-module. A submodule \(N\) of \(M\) is called prime if \(N\neq M\) and for each \(a\in R\), \(m\in M\), then \(am\in N\) implies \(m\in N\) or \(aM\subseteq N\). Hence if \(N\) is a prime submodule of \(M\), then the annihilator \(\mathfrak p\) of \(M/N\) is a prime ideal of \(R\) [ cf. \textit{R. L. McCasland} and \textit{M. E. Moore}, Commun. Algebra 20, No. 6, 1803-1817 (1994; Zbl 0776.13007) and \textit{R. L. McCasland} and \textit{P. F. Smith}, Rocky Mt. J. Math. 23, No. 3, 1041-1062 (1993; Zbl 0814.16017)]. The authors obtain interesting results on the relation between prime submodules of \(M\) and prime ideals of \(R\). They prove that to every prime submodule of \(M\) there corresponds a prime ideal of \(R\) and a set of linear equations of a certain type, and conversely.
reviewed by: Siamak Yassemi
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 1271831
zbMATH Keywords: prime submodules
zbMATH Keywords: prime ideals
zbMATH Keywords: linear equations
author: Yücel Tiraş
author: Abdullah Harmanci
author: Patrick F. Smith
MaRDI profile type: Publication
OpenAlex ID: W2090889127
cites work: A principal ideal theorem analogue for modules over commutative rings
cites work: On the prime radical of a module over a commutative ring
cites work: Prime submodules
cites work: Prime submodules of Noetherian modules
DOI: 10.1006/JABR.1998.7636
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description: scientific article; zbMATH DE number 1271831