Essentially compressible modules and rings

Let R be a ring with identity and let M be a unitary right R-module. Then M is essentially compressible provided M embeds in every essential submodule of M. It is proved that every non-singular essentially compressible module M is isomorphic to a submodule of a free module, and the converse holds in case R is semiprime right Goldie. In case R is a right FBN ring, M is essentially compressible if and only if M is subisomorphic to a direct sum of critical compressible modules. The ring R is right essentially compressible if and only if there exist a positive integer n and prime ideals Pi (1⩽i⩽n) such that P1∩⋯∩Pn=0 and the prime ring R/Pi is right essentially compressible for each 1⩽i⩽n. It follows that a ring R is semiprime right Goldie if and only if R is a right essentially compressible ring with at least one uniform right ideal.