When do maximal submodules force linearity?
type: Item
instance of: scholarly article
title: When do maximal submodules force linearity?
zbMATH Open document ID: 0953.16003
DOI: 10.1016/S0022-4049(98)00041-3
author: P. R. Fuchs
author: C. J. Maxson
published in: Journal of Pure and Applied Algebra
publication date: 1999-11-23T00:00:00Z
review text: All rings \(R\) considered have an identity and are not division rings, while all \(R\)-modules considered are left \(R\)-modules which are unitary, faithful and not cyclic. Maximal submodules of \(R\)-modules are assumed to be non-zero modules and if \(X\) is a subset of an \(R\)-module \(V\) and \(v\in V\), then the notation \((X:v)=\{r\in R:rv\in X\}\) is used. Further, \(\text{Soc}(V)\) denotes the socle, i.e. the sum of all simple submodules, of \(V\). Let \(R\) be a ring and \(M\) a maximal submodule of an \(R\)-module \(V\). Then \(M\) is said to force linearity on \(V\) if every map \(f\colon V\to V\) with \(f(rv)=rf(v)\) for all \(r\in R\) and all \(v\in V\) and \(f(m+n)=f(m)+f(n)\) for all \(m,n\in M\) is an \(R\)-endomorphism of \(V\). First, a rather technical but general characterization of maximal submodules that force linearity is given (Theorem 2.2). For special types of rings this characterization leads to several results such as the following ones. If \(R\) is a local ring and \(V\) is an \(R\)-module, then every maximal submodule forces linearity on \(V\) if \(\text{Soc}(V)=\{0\}\) while otherwise no maximal submodule forces linearity on \(V\) (Theorem 3.1). If \(R\) is a commutative ring, \(V\) is an \(R\)-module, \(M\) a maximal submodule of \(V\), and \(V=M+Rv\) for some \(v\in V-M\), then the following statements are equivalent: (i) \(M\) does not force linearity on \(V\), (ii) there exists \(w\in V\) such that \(\text{Ann}_R(w)=(M:v)\), and there exists \(m\in M\) such that \((Rv:m)\subseteq(M:v)\) (Theorem 3.2). If \(R\) is a commutative ring, \(P\) a prime ideal of \(R\) and \(V\) is an \(R\)-module, then \(V_P\) denotes the localization of \(V\) by the prime \(P\). Suppose \(R\) has the ascending chain condition on annihilator ideals, \(V\) is an \(R\)-module and \(M\) a maximal submodule of \(V\). Then the following statements are equivalent: (i) \(M\) forces linearity on \(V\), (ii) for every prime ideal \(P\) of \(R\), \(M_P\) forces li…
reviewed by: Hans Lausch
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16D10
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16P60
zbMATH DE Number: 1340436
zbMATH Keywords: injectivity
zbMATH Keywords: endomorphisms
zbMATH Keywords: simple submodules
zbMATH Keywords: maximal submodules
zbMATH Keywords: local rings
zbMATH Keywords: localizations
zbMATH Keywords: ascending chain condition on annihilator ideals
MaRDI profile type: Publication
full work available at URL: s0022-4049(98)00041-3
OpenAlex ID: W2008698079
Wikidata QID: Q128133079
description: scientific article; zbMATH DE number 1340436