MODULES WHOSE CLOSED SUBMODULES WITH ESSENTIAL SOCLE ARE DIRECT SUMMANDS

Abstract

We introduce and study CLESS-modules, which subsume two generalizations of extending modules due to P.F. Smith and A. Tercan. A module M will be called a CLESS-module if every closed submodule N of M (in the sense that M/N is non-singular) with essential socle is a direct summand of M. Various properties concerning direct sums of CLESS-modules are established. We show that, over a Dedekind domain, a module is CLESS if and only if its torsion submodule is a direct summand. We also study the behaviour of CLESS-modules under excellent extensions of rings.