A note on a problem due to Zelmanowitz

In this paper we consider a problem due to Zelmanowitz. Specifically, we study under what conditions a uniform compressible module whose nonzero endomorphisms are monomorphisms is critically compressible. We give a positive answer to this problem for the class of nonsingular modules, quasi-projective modules and for modules over rings which are in a certain class of rings which contains at least the commutative rings and the left duo rings.