SS-supplemented modules

Abstract

A module M is called ss-supplemented if every submodule U of M has a supplement V in M such that U(intersection) V is semisimple. It is shown that a finitely generated module M is ss-supplemented if and only if it is supplemented and Rad(M) (submodule) Soc(M). A module M is called strongly local if it is local and Rad(M) (submodule) Soc(M). Any direct sum of strongly local modules is ss-supplemented and coatomic. A ring R is semiperfect and Rad(R) (submodule) Soc( R R) if and only if every left R-module is ss-supplemented if and only if R R is a finite sum of strongly local submodules.