Covers of acts over monoids II

TL;DR

This paper extends the study of flat and other types of covers to acts over monoids, revealing differences from the classical module case and exploring various cover types like free, divisible, and injective.

Contribution

It introduces new results on covers of acts over monoids, expanding the understanding of their properties and differences from module covers.

Findings

Results differ from module case in some instances

Extended the study to free, divisible, torsion free, and injective covers

Provided preliminary results on the 'other' type of cover

Abstract

In 1981 Edgar Enochs conjectured that every module has a flat cover and finally proved this in 2001. Since then a great deal of effort has been spent on studying different types of covers, for example injective and torsion free covers. In 2008, Mahmoudi and Renshaw initiated the study of flat covers of acts over monoids but their definition of cover was slightly different from that of Enochs. Recently, Bailey and Renshaw produced some preliminary results on the `other' type of cover and it is this work that is extended in this paper. We consider free, divisible, torsion free and injective covers and demonstrate that in some cases the results are quite different from the module case.