On covers of acts over monoids with condition $$(PF'')$$ and envelopes of S-acts

Abstract

In 2001, Enochs’ celebrated flat cover conjecture was finally proven, and the proofs have since generated a great deal of interest among researchers. In 2008, Mahmoudi and Renshaw initiated the study of flat covers of acts over monoids which concerned with the coessential epimorphisms. In this paper, we restrict our attention to $$(PF'')$$ -covers (coessential-covers that satisfy Condition $$(PF'')$$ ). We give a necessary and sufficient condition for cyclic act to have a $$(PF'')$$ -cover and give some classes of monoids that all cyclic right S-acts have a Condition $$(PF'')$$ -cover, we show that every weakly pullback flat cover is also $$(PF'')$$ -cover and every $$(PF'')$$ -cover is $$(P')$$ -cover, but the converses are not true. Besides we consider the $$\mathcal {PF''}$$ -cover (Enochs’ notion of a cover) and obtain some properties. Finally, we define and investigate the envelopes of S-acts.