Submodules of direct sums of compressible modules submodules of direct sums of compressible modules

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.