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Let R be a commutative ring, M an R-module and be the endomorphism of M given by right multiplication by . We say that M is weakly-morphic if as R-modules for every a. We study these modules and use them to characterise the rings , where is the right annihilator of M. A kernel-direct or image-direct module M is weakly-morphic if and only if each element of 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 is a field. A finitely generated -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 for some non-negative integers n and k.