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A right R-module M is called coretractable (s-coretractable) if Hom(M/K , M) = 0 for any proper submodule (supplement submodule) K of M. In this article, we continue the study of coretractable modules. Then we study s-coretractable modules. It is shown that this property is not inherited by direct summands and a direct sum of s-coretractable modules may not be s-coretractable. Examples are provided to illustrate and delineate the results.

Abstract. Dedekind domains, Artinian serial rings and right uniserial rings share the following property: Every cyclic right module is a direct sum of uniform modules. We first prove the following improvement of the well-known Osofsky-Smith theorem: A cyclic module with every cyclic subfactor a direct sum of extending modules has finite Goldie dimension. So, rings with the above-mentioned property are precisely rings of the title. Non-commutative rings whose cyclic (right) modules satisfy injectivity-like properties have been studied in the last four decades since the first non-trivial result in this direction was proved by Osofsky ([16] and [17]): She showed that a ring is semisimple Artinian if its cyclic right modules are injective. Klatt and Levy characterised in [12] commutative rings whose factors are self-injective, a noncommutative analogue of which was studied by Ahsan ([1, 2]) and Koehler ([13]), where they considered rings whose cyclic right (equivalently left) modules are quasi-injective, namely qc-rings.