Abstract
References
- D. D. Anderson, T. Arabaci, U. Tekir and S. Koc, On S-multiplication modules, Comm. Algebra, 48(8) (2020), 3398-3407.
- A. Barnard, Multiplication modules, J. Algebra, 71(1) (1981), 174-178.
- S. Baupradist and S. Asawasamrit, On fully-M-cyclic modules, J. Math. Res., 3(2) (2011), 23-26.
- S. Baupradist and S. Asawasamrit, GW-principally injective modules and pseudo-GW-principally injective modules, Southeast Asian Bull. Math., 42 (2018), 521-529.
- S. Baupradist, H. D. Hai and N. V. Sanh, On pseudo-p-injectivity, Southeast Asian Bull. Math., 35(1) (2011), 21-27.
- S. Baupradist, H. D. Hai and N. V. Sanh, A general form of pseudo-p-injectivity, Southeast Asian Bull. Math., 35 (2011), 927-933.
- A. Haghany and M. R. Vedadi, Modules whose injective endomorphisms are essential, J. Algebra, 243(2) (2001), 765-779.
- V. Kumar, A. J. Gupta, B. M. Pandeya and M. K. Patel, M-sp-injective modules, Asian-Eur. J. Math., 5(1) (2012), 1250005 (11 pp).
- S. H. Mohamed and B. J. Muller, Continuous and Discrete Modules, London Math. Soc. Lecture Note Series, 147, Cambridge Univ. Press, Cambridge, 1990.
- W. K. Nicholson and M. F. Yousif, Principally injective rings, J. Algebra, 174 (1995), 77-93.
- M. K. Patel and S. Chase, FI-semi-injective modules, Palest. J. Math., 11(1) (2022), 182-190.
- M. K. Patel, B. M. Pandeya, A. J. Gupta and V. Kumar, Quasi principally injective modules, Int. J. Algebra, 4 (2010), 1255-1259.
- T. C. Quynh and N. V. Sanh, On quasi pseudo-GP-injective rings and modules, Bull. Malays. Math. Sci. Soc., 37(2) (2014), 321-332.
- N. V. Sanh and K. P. Shum, Endomorphism rings of quasi principally injective modules, Comm. Algebra, 29(4) (2001), 1437-1443.
- N. V. Sanh, K. P. Shum, S. Dhompongsa and S.Wongwai, On quasi-principally injective modules, Algebra Colloq., 6(3) (1999), 269-276.
- W. M. Xue, On Morita duality, Bull. Austral. Math. Soc., 49(1) (1994), 35-45.
- Z. Zhu, Pseudo QP-injective modules and generalized pseudo QP-injective modules, Int. Electron. J. Algebra, 14 (2013), 32-43.
Abstract
In this paper, we introduce the notion of $S$-$M$-cyclic submodules, which is a generalization of the notion of $M$-cyclic submodules. Let $M, N$ be right $R$-modules and $S$ be a multiplicatively closed subset of a ring $R$. A submodule $A$ of $N$ is said to be an $S$-$M$-cyclic submodule, if there exist $s\in S$ and $f \in Hom_R(M,N)$ such that $As \subseteq f(M) \subseteq A$. Besides giving many properties of $S$-$M$-cyclic submodules, we generalize some results on $M$-cyclic submodules to $S$-$M$-cyclic submodules. Furthermore, we generalize some properties of principally injective modules and pseudo-principally injective modules to $S$-principally injective modules and $S$-pseudo-principally injective modules, respectively. We study the transfer of this notion to various contexts of these modules.
Keywords
$M$-cyclic submodule $S$-$M$-cyclic submodule $M$-principally injective module $S$-$M$-principally injective module pseudo-$M$-principally injective module $S$-pseudo-$M$-principally injective module
References
- D. D. Anderson, T. Arabaci, U. Tekir and S. Koc, On S-multiplication modules, Comm. Algebra, 48(8) (2020), 3398-3407.
- A. Barnard, Multiplication modules, J. Algebra, 71(1) (1981), 174-178.
- S. Baupradist and S. Asawasamrit, On fully-M-cyclic modules, J. Math. Res., 3(2) (2011), 23-26.
- S. Baupradist and S. Asawasamrit, GW-principally injective modules and pseudo-GW-principally injective modules, Southeast Asian Bull. Math., 42 (2018), 521-529.
- S. Baupradist, H. D. Hai and N. V. Sanh, On pseudo-p-injectivity, Southeast Asian Bull. Math., 35(1) (2011), 21-27.
- S. Baupradist, H. D. Hai and N. V. Sanh, A general form of pseudo-p-injectivity, Southeast Asian Bull. Math., 35 (2011), 927-933.
- A. Haghany and M. R. Vedadi, Modules whose injective endomorphisms are essential, J. Algebra, 243(2) (2001), 765-779.
- V. Kumar, A. J. Gupta, B. M. Pandeya and M. K. Patel, M-sp-injective modules, Asian-Eur. J. Math., 5(1) (2012), 1250005 (11 pp).
- S. H. Mohamed and B. J. Muller, Continuous and Discrete Modules, London Math. Soc. Lecture Note Series, 147, Cambridge Univ. Press, Cambridge, 1990.
- W. K. Nicholson and M. F. Yousif, Principally injective rings, J. Algebra, 174 (1995), 77-93.
- M. K. Patel and S. Chase, FI-semi-injective modules, Palest. J. Math., 11(1) (2022), 182-190.
- M. K. Patel, B. M. Pandeya, A. J. Gupta and V. Kumar, Quasi principally injective modules, Int. J. Algebra, 4 (2010), 1255-1259.
- T. C. Quynh and N. V. Sanh, On quasi pseudo-GP-injective rings and modules, Bull. Malays. Math. Sci. Soc., 37(2) (2014), 321-332.
- N. V. Sanh and K. P. Shum, Endomorphism rings of quasi principally injective modules, Comm. Algebra, 29(4) (2001), 1437-1443.
- N. V. Sanh, K. P. Shum, S. Dhompongsa and S.Wongwai, On quasi-principally injective modules, Algebra Colloq., 6(3) (1999), 269-276.
- W. M. Xue, On Morita duality, Bull. Austral. Math. Soc., 49(1) (1994), 35-45.
- Z. Zhu, Pseudo QP-injective modules and generalized pseudo QP-injective modules, Int. Electron. J. Algebra, 14 (2013), 32-43.
Details
| Primary Language | English |
|---|---|
| Subjects | Algebra and Number Theory |
| Journal Section | Articles |
| Authors | |
| Early Pub Date | May 7, 2024 |
| Publication Date | |
| Submission Date | January 9, 2024 |
| Acceptance Date | April 2, 2024 |
| Published in Issue | Year 2024 Early Access |