Generators for the semimodule of varieties of a free module
type: Item
instance of: scholarly article
title: Generators for the semimodule of varieties of a free module
zbMATH Open document ID: 0980.13007
author: Roy L. McCasland
author: Patrick F. Smith
author: M. E. Moore
published in: Rocky Mountain Journal of Mathematics
publication date: 2002-03-10T00:00:00Z
full work available at URL: CONT29-4.html
review text: Let \(R\) be a commutative ring with identity, and let \(M\) be an \(R\)-module. For a submodule \(N\) of \(M\) let \((N:M)= \{r\in R\mid rM \subseteq N\}\). The submodule \(N\) is called prime if \((N:M)\) is a prime ideal of \(R\) and the \(R/(N:M)\)-module \(M/N\) is torsion-free. In analogy to the spectrum of a ring one can define the set \(\text{Spec}(M)\) to be the collection of all prime submodules of \(M\). Let \(V(N)\), the variety of \(N\), be the collection of all prime submodules containing \(N\). The collection of all such varieties is denoted by \(\zeta(M)\). It can be shown that \(\zeta(M)\) is a semi-module over the semi-ring \(\zeta(R)\). This paper investigates the question when \(\zeta(M)\) is finitely generated as a semi-module over the semi-ring \(\zeta(R)\). The main result is that, if \(R\) is an infinite Noetherian UFD and \(M\) is a finitely generated \(R\)-module, then \(\zeta(M)\) is finitely generated if and only if \(M\cong R\oplus T\), where \(T\) is a finite \(R\)-module.
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 54B35
Mathematics Subject Classification ID: 13F15
zbMATH DE Number: 1455479
zbMATH Keywords: Zariski-finite modules
zbMATH Keywords: variety of submodule
zbMATH Keywords: prime submodules
reviewed by: Reinhard Laubenbacher
MaRDI profile type: Publication
OpenAlex ID: W2037967268
cites work: Q3905426
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DOI: 10.1216/RMJM/1181070416
description: scientific article; zbMATH DE number 1455479