Ring With All Finitely Generated Modules Retractable

Several characterizations of a ring R is given for which any non-zero finitely generated module M is retractable in the sense that HomR(M,N) is non-zero whenever N is a non-zero submodule of M. Such rings are called finite retractable. It is shown that any ring being Morita equivalent to a commutative ring is finite retractable. Also, if the commutative ring is semi-Artinian then any non-zero module is retractable. The class of finite retractable rings is shown to be closed under Morita equivalence and finite direct products. Moreover, for a finite retractable ring R which is a right order in a ring Q, it is shown that Q is also finite retractable.