Covers of acts over monoids

Abstract

Since they were first defined in the 1950's, projective covers (the dual of injective envelopes) have proved to be an important tool in module theory, and indeed in many other areas of abstract algebra. An attempt to generalise the concept led to the introduction of covers with respect to other classes of modules, for example, injective covers, torsion-free covers and at covers. The at cover conjecture (now a Theorem) is of particular importance, it says that every module over every ring has a at cover. This has led to surprising results in cohomological studies of certain categories. Given a general class of objects X, an X-cover of an object A can be thought of a the 'best approximation' of A by an object from X. In a certain sense, it behaves like an adjoint to the inclusion functor. In this thesis we attempt to initiate the study of different types of covers for the category of acts over a monoid. We give some necessary and sufficient conditions for the existence of X covers for a general class X of acts, and apply these results to specific classes. Some results include, every S act has a strongly at cover if S satisfies Condition (A), every S-act has a torsion free cover if S is cancellative, and every S-act has a divisible cover if and only if S has a divisible ideal. We also consider the important concept of purity for the category of acts. Giving some new characterisations and results for pure monomorphisms and pure epimorphisms.

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ORCID for J.H. Renshaw: orcid.org/0000-0002-5571-8007

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