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arXiv · 2210.04970

On Hopfian(co-Hopfian) and Fitting S-acts (I)

Mohammad Roueentan·Roghaieh Khosravi
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Abstract

The main purpose of the present work is an investigation of the notions Hopfian (co-Hopfian) acts whose their surjective (injective) endomorphisms are isomorphisms. While we investigate conditions that are relevant to these classes of acts, their interrelationship with some other concepts for example quasi-injective and Dedekind-finite acts is studied. Using Hopfian and co-Hopfian concepts, several conditions are given for a quasi-injective act to be Dedekind-finite. Moreover we bring out some properties of strongly Hopfian and strongly co-Hopfian $S$-acts. Ultimately we introduce and study the concept of Fitting acts and over a monoid $S$, some equivalent conditions are found to have all its finitely generated (cyclic) acts Fitting. It is shown that an $S$-act is Fitting if and only if it is both strongly Hopfian and strongly co-Hopfian.

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Mohammad Roueentan, Roghaieh Khosravi. 2022-10-10. On Hopfian(co-Hopfian) and Fitting S-acts (I). https://arxiv.org/abs/2210.04970

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