Morphic Modules

A module M is called morphic if M/M α ≅ ker(α) for all endomorphisms α in end(M), and a ring R is called a left morphic ring if RR is a morphic module. We consider the open question when the matrix ring Mn(R) is left morphic by relating it to when Rn is morphic as a left R-module. More generally, we investigate when M being morphic implies that end(M) is left morphic, and conversely. Finally, we relate the morphic condition to internal cancellation in the module.