When are homogeneous functions linear? A lattice point of view.
Throughout, \(R\) is a ring with identity and all \(R\)-modules are unital left modules. If \(V\) and \(W\) are \(R\)-modules, a function \(f\colon V\to W\) is said to be \(R\)-homogeneous if \(f(rv)=rf(v)\) for all \(r\in R\) and \(v\in V\). Following terminology introduced by \textit{U. Albrecht} and the reviewer [Proc. Am. Math. Soc. 123, No. 8, 2381-2389 (1995; Zbl 0839.16024)], \(V\) is a ray if for every \(R\)-module \(W\) and every \(R\)-homogeneous \(f\colon V\to W\), the map \(f\) is additive, i.e. \(f\) is an \(R\)-module homomorphism. The authors study relationships between the submodule lattice of \(V\) and the condition that \(V\) be a ray. Their results are applied to further determine the structure of those rings \(R\) for which every \(R\)-module is a ray. A characterization of all semisimple modules which are rays is obtained.
scientific article; zbMATH DE number 2174566
scientific article