Modules Whose Submodules Are Essentially Embedded in Direct Summands

Abstract

A module M is said to satisfy the C-12 condition if every submodule of M is essentially embedded in a direct summand of M. It is known that the C-11 ( and hence also C-1) condition implies the C-12 condition. We show that the class of C-12-modules is closed under direct sums and also essential extensions whenever any module in the class is relative injective with respect to its essential extensions. We prove that if M is a C-12(+)-module with cancellable socle and satisfies ascending chain ( respectively, descending chain) condition on essential submodules, then M is a direct sum of a semisimple and a Noetherian ( respectively, Artinian) submodules. Moreover, a C-12-module with cancellable socle is shown to be a direct sum of a module with essential socle and a module with zero socle. An example is constructed to show that the reverse of the last result do not hold.