Submodules of Direct Sums of Compressible Modules

Abstract

Let R be a ring. A right R-module M is called “essentially compressible” if it embeds in each of its essential submodules. Also a module X R is called “completely essentially compressible” if every submodule of X R is an essentially compressible R-module. In this aricle, it is shown that a right R-module M embeds in a direct sum of compressible right R-modules if and only if M R is essentially compressible and every nonzero essentially compressible submodule of M R contains a compressible submodule. Every essentially compressible R-module is shown to be retractable. Moreover, if either R R has Krull dimension, or R is Morita equivalent to a right duo ring, then a right R-module embeds in a direct sum of compressible right R-modules if and only if it is completely essentially compressible.