Generalizations of Rad-Supplemented Modules

Abstract

Let R be an associative ring with identity. We introduce the notion of semi-tau-supplemented modules, which is adapted from srs-modules, for a preradical tau on R-Mod. We provide basic properties of these modules. In particular, we study the objects of R-Mod for tau = Rad. We show that the class of semi-tau-supplemented modules is closed under finite sums and factor modules. We prove that, for an idempotent preradical tau on R-Mod, a module M is semi-tau-supplemented if and only if it is tau-supplemented. For tau = Rad, over a local ring every left module is semi-Rad-supplemented. We also prove that a commutative semilocal ring whose semi-Rad-supplemented modules are a direct sum of w-local left modules is an artinian principal ideal ring.