Modules whose submodules are essentially embedded in direct summands

A module M is said to satisfy the C₁₂ condition if every submodule of M is essentially embedded in a direct summand of M. It is known that the C₁₁ (and hence also C₁) condition implies the C₁₂ condition. We show that the class of C₁₂-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 [image omitted]-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 C₁₂-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.