Abstract
A module M is called dual-square-free (DSF) if M has no proper submodules A and B with and The class of DSF-modules is closed under direct summands and homomorphic images, and a module M is distributive iff every submodule of M is dual-square-free. In this article we consider certain classes of rings R over which is a DSF-module. For example, if R is a right hereditary ring such that is DSF, then R is right noetherian, right distributive, every right ideal of R is two-sided, and every subfactor of R-R is quasi-continuous. Also, if R is semilocal and is a DSF-module with small radical, then R is basic, semiperfect and right self-injective. As an immediate application, if R is a right perfect ring such that is DSF as a right R-module then R is right Artinian. If in addition we assume that either or is a DSF-module then the ring R becomes quasi-Frobenius.