(r,t)-injectivity in the category $S$-Act

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran.

10.29252/cgasa.11.1.169

Abstract

In this paper, we show that injectivity with respect to the class $\mathcal{D}$  of dense monomorphisms of an idempotent and weakly hereditary closure operator of an arbitrary category  well-behaves. Indeed, if $\mathcal{M}$ is a subclass of monomorphisms, $\mathcal{M}\cap \mathcal{D}$-injectivity  well-behaves. We also introduce the notion of $(r,t)$-injectivity in the category {\bf S-Act}, where $r$ and $t$ are Hoehnke radicals, and discuss whether this kind of injectivity well-behaves.

Keywords


[1] Adamek, J., Herrlich, H., and Strecker, G.E., “Abstract and Concrete Categories”, John Wiley and Sons, 1990.
[2] Balbes, R., Projective and injective distributive lattices, Pacific. J. Math. 21(3) (1967), 405-420.
[3] Banaschewski, B., Injectivity and essential extensions in equational classes of algebras, Queen’s Papers in Pure and Applied Mathematics 25 (1970), 131-147.
[4] Beachy, J., A generalization of injectivity, Pacific J. Math. 41(2) (1972), 313-327.
[5] Bican, L., Preradicals and injectivity, Pacific J. Math. 56(2) (1975), 367-372.
[6] Burris, S. and Sankapanavar, H.P., “A Course in Universal Algebra”, Springer-Verlag, 1981.
[7] Clementino, M., Dikranjan, D., and Tholen, W., Torsion theories and radicals in normal categories, J. Algebra 305(1) (2006), 98-129.
[8] Crivei, S., “Injective Modules Relative to Torsion Theories” Editura Fundaµiei pentru Studii Europene, 2004.
[9] Dikranjan, D., and Tholen, W., “Categorical Structure of Closure Operators: with Applications to Topology, Algebra and Discrete Mathematics”, Kluwer Academic Publishers, 1995.
[10] Ebrahimi, M.M. and Barzegar, H., Sequentially pure monomorphisms of acts over semigroups, Eur. J. Pure Appl. Math. 1(4) (2008), 41-55.
[11] Ebrahimi, M.M., Haddadi, M., and Mahmoudi, M., Injectivity in a category: an overview on well behavior theorems, Algebras Groups and Geometries 26(4) (2009), 451-472.
[12] Ebrahimi, M.M., Haddadi, M., and Mahmoudi, M., Injectivity in a category: an overview on smallness conditions, Categ. Gen. Algebr. Struct. Appl. 2(1) (2015), 83-112.
[13] Gould, V., The characterisation of monoids by properties of their S-systems, Semigroup forum 32 (1985), 251-265.
[14] Haddadi, M. and Ebrahimi, M.M., A radical extension of the category of S-sets, Bull. Iranian Math. Soc. 43(5) (2017), 1153-1163.
[15] Haddadi, M. and Sheykholislami, S.M.N., Radical-injectivy in the category S-act, arXiv:1806.07077v1, 2018.
[16] Jirásko, J., Generalized injectivity, Comment. Math. Univ. Carolin. 16(4) (1975), 621-636.
[17] Kilp, M., Knauer, U., and Mikhalev, A.V., “Monoids, Acts and Categories”, Walter de Gruyter, 2000.
[18] Maranda, J.M., Injective structures, Trans. Amer. Math. Soc. 110(1) (1964), 98-135.
[19] Mehdi, A.R., On l-injective modules, arXiv:1501.02491v2, 2017.
[20] Rosick`y, J., On the uniqueness of cellular injectives, arXiv:1702.08684v2, 2018.
[21] Tholen, W., Injectivity versus exponentiability, Cah. Topol. Géom. Différ. Catég 49(3) (2008), 228-240.
[22] Wiegandt, R., Radical and torsion theory for acts, Semigroup Forum 72 (2006), 312-328.