Hacettepe Journal of Mathematics and Statistics 2651-477X 2651-477X Hacettepe University 10.15672/hujms.552216 Mathematical Sciences Matematik On covers of acts over monoids with Condition $(P')$ https://orcid.org/0000-0001-7795-2809 Irannezhad Setareh Semnan University https://orcid.org/0000-0002-0195-3079 Madanshekaf Ali Semnan University 02 06 2020 49 1 352 361 07 01 2017 12 11 2018 Copyright © 2002, Hacettepe Journal of Mathematics and Statistics 2002 Hacettepe Journal of Mathematics and Statistics

In this paper we consider two different definitions of cover, one of them is Enochs' notion of a cover and the other is the one that initiated by Mahmoudi and Renshaw which concerned with the coessential epimorphisms. We show that these definitions are not equivalent in our 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 $(P')$-cover and a sufficient condition for every act to have a $\mathcal{P'}$-cover (Enochs' $\mathcal{P'}$-cover where $\mathcal{P'}$ is the class of $S$-acts satisfying Condition $(P')$). We also obtain numerous classes of monoids over which indecomposable acts satisfying Condition $(P')$ are locally cyclic.

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