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Covers of Acts Over Monoids and Pure Epimorphisms

Published online by Cambridge University Press:  16 April 2014

Alex Bailey  and
James H. Renshaw
Alex Bailey
Affiliation:
Department of Mathematics, University of Southampton, Southampton SO17 1BJ, UK, (alex.bailey@soton.ac.uk; j.h.renshaw@maths.soton.ac.uk)
James H. Renshaw
Affiliation:
Department of Mathematics, University of Southampton, Southampton SO17 1BJ, UK, (alex.bailey@soton.ac.uk; j.h.renshaw@maths.soton.ac.uk)
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Abstract

In 2001, Enochs's celebrated flat cover conjecture was finally proven, and the proofs (two different proofs were presented in the same paper) have since generated a great deal of interest among researchers. The results have been recast in a number of other categories and, in particular, for additive categories. In 2008, Mahmoudi and Renshaw considered a similar problem for acts over monoids but used a slightly different definition of cover. They proved that, in general, their definition was not equivalent to that of Enochs, except in the projective case, and left open a number of questions regarding the ‘other’ definition. This ‘other’ definition is the subject of the present paper and we attempt to emulate some of Enochs's work for the category of acts over monoids, and concentrate, in the main, on strongly flat acts. We hope to extend this work to other classes of acts, such as injective, torsion free, divisible and free, in a future report.

Keywords

monoidsactsstrongly flatcoverspure epimorphismscolimitsPrimary 20M50
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 3 , October 2014 , pp. 589 - 617
Copyright
Copyright © Edinburgh Mathematical Society 2014 

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References

1.Ähsan, J. and Liu, Z., A homological approach to the theory of monoids (Science Press, Beijing, 2008).Google Scholar
2.Bican, L., Some properties of precovers and covers, Alg. Representat. Theory 12 (2009), 311–318.Google Scholar
3.Bican, L. and Torrecillas, B., On covers, J. Alg. 236 (2001), 645–650.Google Scholar
4.Bican, L., El Bashir, R. and Enochs, E., All modules have flat covers, Bull. Lond. Math. Soc. 33 (2001), 385–390.Google Scholar
5.Bridge, P., Essentially algebraic theories and localizations in toposes and abelian categories, PhD Thesis, University of Manchester (2012).Google Scholar
6.El Bashir, R., Covers and directed colimits, Alg. Representat. Theory 7 (2004), 423–430.Google Scholar
7.Fountain, J., Perfect semigroups, Proc. Edinb. Math. Soc. 20 (1969), 87–93.Google Scholar
8.Howie, J. M., Fundamentals of semigroup theory, London Mathematical Society Monographs, Volume 11 (Clarendon, Oxford, 1995).Google Scholar
9.Isbell, J. R., Perfect monoids, Semigroup Forum 2 (1971), 95–118.Google Scholar
10.Khosravi, R., Ershad, M. and Sedaghatjoo, M., Strongly flat and condition (P) covers of acts over monoids, Commun. Alg. 38 (2010), 4520–4530.Google Scholar
11.Kilp, M., Knauer, U. and Mikhalev, A. V., Monoids, acts and categories, De Gruyter Expositions in Mathematics, Volume 29 (Walter de Gruyter, Berlin, 2000).Google Scholar
12.Lazard, D., Autour de la platitude, Bull. Soc. Math. France 97 (1969), 81–128.Google Scholar
13.Mahmoudi, M. and Renshaw, J., On covers of cyclic acts over monoids, Semigroup Forum 77 (2008), 325–338.Google Scholar
14.Normak, P., On noetherian and finitely presented polygons, Acta Comment. Univers. Tartuensis 431 (1977), 37–46 (Engl. transl. available at http://sites.google.com/site/cdhollings/translations).Google Scholar
15.Normak, P., On equalizer-flat and pullback-flat acts, Semigroup Forum 36 (1987), 293–313.CrossRefGoogle Scholar
16.Renshaw, J., Extension and amalgamation in monoids, semigroups and rings, PhD Thesis, University of St andrews (1985).Google Scholar
17.Renshaw, J., Extension and amalgamation in monoids and semigroups, Proc. Lond. Math. Soc. 52(3) (1986), 119–141.Google Scholar
18.Renshaw, J., Perfect amalgamation bases, J. Alg. 141(1) (1991), 78–92.CrossRefGoogle Scholar
19.Renshaw, J., Monoids for which condition (P) acts are projective, Semigroup Forum 61 (2000), 46–56.Google Scholar
20.Renshaw, J., Stability and flatness in acts over monoids, Colloq. Math. 92(2) (2002), 267–293.Google Scholar
21.Silva, P. V. and Steinberg, B., A geometric characterization of automatic monoids, Q. J. Math. 55(3) (2004), 333–356.Google Scholar
22.Stenstrom, B., Flatness and localisation over monoids, Math. Nachr. 48 (1971), 315–334.Google Scholar
23.Xu, J., Flat covers of modules, Lecture Notes in Mathematics, Volume 1634 (Springer, 1996).Google Scholar