Modules for which homogeneous maps are linear
For a ring \(R\) and an \(R\)-module \(V\), the near-ring of homogeneous maps \({\mathcal M}_R(V)\) is the set of maps \(\{f\colon V\to V;\;f(rv)=rf(v)\) for all \(r\in R\) and \(v\in V\}\), with pointwise addition and composition of functions. These have been studied for some time (\textit{C. J. Maxson} and \textit{K. C. Smith} [Proc. Am. Math. Soc. 80, 189-195 (1980; Zbl 0454.16022)] and \textit{C. J. Maxson} and \textit{A. P. J. van der Walt} [J. Aust. Math. Soc., Ser. A 50, No. 2, 279-296 (1991; Zbl 0731.16029)]). The author considers the case when \(R\) is a commutative Noetherian ring. Assume that \(V\) is finitely generated. He show that \({\mathcal M}_R(V)=\text{End}_R(V)\) if and only if \(V_P\) is \(R_P\)-cyclic for all \(P\) in \(\text{Max-Ass }V\). \(P\) belongs to \(\text{Ass }V\) if it is a prime ideal of the form \(\text{Ann}_R(v)\), and \(\text{Max-Ass }V\) is the set of maximal elements in this set. The other main result is that \({\mathcal M}_R(V)=\text{End}_R(V)\) for all uniform modules \(V\) if and only if the maximal ideal of \(R_P\) is principal for each prime ideal \(P\) of \(R\).
scientific article; zbMATH DE number 1455483
scientific article