Let R be a ring, M be a nonzero R-module and F: \\sigma[M] * \\sigma[M] ------ Mod-Z be a bifunctor. The F-reversibility of M is defined by F(X; Y) = 0 implies F(Y;X) = 0 for all non-zero X, Y in \\sigma[M]. Hom (resp. Rej)-reversibility of M is characterized in different ways. Among other things, it is shown that R is Hom-reversible if and only if R = such that each Ri is a perfect ring with a unique simple module (up to isomorphism). In particular, for a duo ring, the concepts of perfectness and Hom-reversibility coincide. Also, we carry out a study of rings R for which Hom(M;N) is nonzero for all nonzero submodule N of M. Such rings are called retractable. For a retractable ring, Artinian condition and having Krull dimension are equivalent. Furthermore, a right Artinian ring in which prime ideals commute is precisely a right Noetherian retractable ring. Retractable rings are characterized in several ways. They form a class of rings that properly lies between the class of pseudo-Frobenius rings, and the class of max divisible rings for which the converse of Schur’s lemma holds. For several types of rings, including commutative rings, retractability is equivalent to semi-Artinian condition. We show that a Kothe ring R is an Artinian principal ideal ring if and only if it is a certain retractable ring, and determine when R is retractable.