Characterizing cogenerating and finitely cogenerated $S$-acts

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Sciences, Fasa University, Fasa, Iran.

2 Department of Mathematics, Shaanxi University of Science and Technology, Xi$'$an, Shaanxi, P.R. China.

3 College of Engineering, Lamerd Higher Education Center, Shiraz University of Technology, Lamerd, Iran.

10.48308/cgasa.2024.235961.1497

Abstract

In this paper, we introduce cogenerating classes of $S$-acts as those that can be used to cogenerate $S$-acts in an appropriate sense. Next, finitely cogenerated $S$-acts are characterized by the property that their socle is finitely cogenerated and large in the $S$-act. Further, we investigate the $S$-acts cogenerating $S_S$, or  generating the injective envelope $E(S)$ of $S_S$.  This leads us to introduce the classes of  cofaithful and subgenerator $S$-acts as the
dual notions of faithful $S$-acts, which  lie strictly between the classes of generator and faithful $S$-acts. Ultimately, we study relations between the cogenerating classes, finitely cogenerated $S$-acts, and the recently introduced new classes of $S$-acts.

Keywords

Main Subjects

References

[1] Abbas, M.S. and Baanoon, H.R., On fully stable acts, Ital. J. Pure Appl. Math. 34 (2015), 389-396.
[2] Anderson, F., Fuller, K., “Rings and Categories of Modules”, Springer-Verlag, 1992.
[3] Barzegar, H., Essentiality in the category of S-acts, Eur. J. Pure Appl. Math. 9(1) (2016), 19-26.
[4] Davvaz and B., Nazemian, Z., Chain conditions on commutative monoids, Semigroup Forum 100 (2020), 732–742.
[5] Estaji, A.A and Shabani, M., A note on multiplication S-acts, Far East J. Math. Sci. 36(2) (2010), 133-150.
[6] Feller, E.H. and Gantos, R.L., Indecomposable and injective S-acts with zero, Math. Nachr. 41 (1969), 37-48.
[7] Gr¨atzer, G., “Universal Algebra”, D. Van Nostrand, Princeton et al., 1968.
[8] Herrlich, H. and Strecker, G., “Category Theory”, Allyn and Bacon, Boston, 1973.
[9] Kilp, M., Knauer, U., and Mikhalev, A., “Monoids, Acts and Categories”, W. de Gruyter, 2000.
[10] Kilp, M. and Knauer, U., On torsionless and dense acts, Semigroup Forum 63 (2001), 396-414.
[11] Khosravi, R. and Roueentan, M., Co-uniform and hollow S-acts over monoids, Commun. Korean Math. Soc. 37(2) (2022), 347–358.
[12] Knauer, U. and Normak, P., Morita duality for monoids , Semigroup Forum 40 (1973), 39-57.
[13] Moghaddasi, G.H., Cogenerator and subdirectly ireducible in the category of S-posets, J. Algebr. Sys. 2(2) (2014), 137-146.
[14] Normak, P., Cogenerator acts, In: A. Cherubini, C. Bonzini, and C. Tibiletti, “Semigroups, Algebraic Theory and Applications to Formal Languages and Codes”, World Scientific, 1993, 191-198.
[15] Sedaghatjoo, M., A correction on the paper: Characterization of monoids by properties of generators, Comm. Algebra 41 (2013), 4662-4666.
[16] Wisbauer, R., “Foundations of Module and Ring Theory”, Gordon and Breach Science Publishers, Reading, 1991.
Categories and General Algebraic Structures with Applications

Articles in Press, Accepted Manuscript
Available Online from 15 September 2024

Files

Share

How to cite

Statistics