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Characterization of monoids by torsion-free, flat, projective, and free acts

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Abstract

A monoidS is susceptible to having properties bearing upon all right acts overS such as: torsion freeness, flatness, projectiveness, freeness. The purpose of this note is to find necessary and sufficient conditions on a monoidS in order that, for example, all flat rightS-acts are free. We do this for all meaningful variants of such conditions and are able, in conjunction with the results of Skornjakov [8], Kilp [5] and Fountain [3], to describe the corresponding monoids, except in the case “all torsion free acts are flat”, where we have only some necessary condition. We mention in passing that homological classification of monoids has been discussed by several authors [3, 4, 5, 8].

In the following,S will always stand for a monoid. A rightS-act is a setA on whichS acts unitarily from the right in the usual way, that is to saya(rs) = (ar)s, a1 =a (a εA,r,s εS) where 1 denotes the identity ofS.

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  1. Ulrich Knauer & Mario Petrich

    Present address: FB 4 Universität Oldenburg Ammerländer Heerstr, 67-99, 2900, Oldenburg, FR Germany

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    1. Ulrich Knauer
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    2. Mario Petrich
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    Knauer, U., Petrich, M. Characterization of monoids by torsion-free, flat, projective, and free acts. Arch. Math 36, 289–294 (1981). https://doi.org/10.1007/BF01223703

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    • DOI: https://doi.org/10.1007/BF01223703

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