Weakly-morphic modules

Let R be a commutative ring, M an R-module and phi(alpha) be the endomorphism of M given by right multiplication by a is an element of R. We say that M is weakly-morphic if M/phi(alpha),(M) similar or equal to ker(phi(alpha)) as R-modules for every alpha. We study these modules and use them to characterise the rings R/Ann(R)(M), where Ann(R)(M) is the right annihilator of M. A kernel-direct or image-direct module M is weakly-morphic if and only if each element of R/Ann(R)(M) is regular as an endomorphism element of M. If M is a weakly-morphic module over an integral domain R, then M is torsion-free if and only if it is divisible if and only if R/Ann(R)(M) is a field. A finitely generated Z-module is weakly-morphic if and only if it is finite; and it is morphic if and only if it is weakly-morphic and each of its primary components is of the form (Z(p)k)(n) for some non-negative integers n and k.