Rings of homogeneous functions
type: Item
instance of: scholarly article
title: Rings of homogeneous functions
zbMATH Open document ID: 0894.16022
DOI: 10.1016/S0022-4049(96)00098-9
author: A. B. van der Merwe
author: C. J. Maxson
published in: Journal of Pure and Applied Algebra
publication date: 1998-03-24T00:00:00Z
review text: Let \(R\) be a ring with identity and let \(V\) be a unital \(R\)-module. A selfmap \(f\) of \(V\) is homogeneous if \(f(rv)=rf(v)\) for all \(r\in R\) and all \(v\in V\). The collection \(M_R(V)\) of all homogeneous selfmaps of \(V\) is a near-ring under pointwise addition and composition and is referred to as the centralizer near-ring determined by \((R,V)\). The authors determine when this near-ring has various properties. For example, they show that if \(R\) is a commutative ring, then \(M_R(V)\) cannot be a near-field without being a field. They show that, in many instances, if \(M_R(V)\) is a ring then it must be commutative. In particular, this is the case whenever \(V\) is finitely generated. However, they also provide an example of an injective \(R\)-module \(V\) such that \(M_R(V)\) is a noncommutative ring. They go on to determine, for a large number of \(R\)-modules \(V\), precisely when \(M_R(V)\) is a ring. For example, they show that if \(V\) is a finitely generated injective module or if \(R\) is Artianian and \(V\) is injective then \(M_R(V)\) is a ring if and only if \(Rx\cap Ry=\{0\}\) for \(x,y\in V\backslash\{0\}\) implies \(\text{Ann}_R(x)\nsubseteq\text{Ann}_R(y)\).
Mathematics Subject Classification ID: 16Y30
Mathematics Subject Classification ID: 13C11
zbMATH DE Number: 1121653
zbMATH Keywords: rings
zbMATH Keywords: homogeneous selfmaps
zbMATH Keywords: centralizer near-rings
zbMATH Keywords: near-fields
zbMATH Keywords: finitely generated injective modules
reviewed by: jun. Kenneth D. Magill
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1121653