Rings of homogeneous functions
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 \(\text{Ann}_R(x)\nsubseteq\text{Ann}_R(y)\).
scientific article; zbMATH DE number 1121653
scientific article