On covers of acts over monoidswith Condition(P′)

Abstract

In this paper we consider two different definitions of cover, one of them is Enochs’ notion ofa cover and the other is the one that initiated by Mahmoudi and Renshaw which concernedwith the coessential epimorphisms. We show that these definitions are not equivalent inour case and restrict our attention to(P′)-covers (coessential-covers that satisfy Condition(P′)). We give a necessary and sufficient condition for a cyclic act to have a(P′)-coverand a sufficient condition for every act to have aP′-cover (Enochs’P′-cover whereP′isthe class ofS-acts satisfying Condition(P′)). We also obtain numerous classes of monoidsover which indecomposable acts satisfying Condition(P′)are locally cyclic.