On Almost ‎$‎S‎$‎-prime submodules

Document Type : Research Paper

Authors

1 Department of Mathematics, Payame Noor University, Tehran, Iran.

2 Department of Mathematics, Payame Noor University, Tehran, Iran

Abstract

‎‎Let $R$ be a commutative ring with non-zero identity, $S\subseteq R$ be a multiplicatively closed subset of $R$ and let $M$ be an $R$-module.  A submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ is said to be almost $S$-prime, if there exists an $s\in S$ such that whenever $rm\in N-(N:_{R}M)N$, then $sm\in N$ or $sr\in (N:_{R}M)$ for each $r\in R$, $m\in M$. The aim of this paper is to introduce and investigate some properties of the notion of almost $S$-prime submodules, especially in multiplication modules. Moreover, we investigate the behaviour of this structure under module homomorphisms, localizations, quotient modules, Cartesian product. Finally, we state two kinds of submodules of the amalgamation module along an ideal and investigate conditions under which they are almost $S$-prime.

Keywords

References

1. M. M. Ali, Residual submodules of multiplication modules, Beit. Algebra Geom.,43 (2007), 321-343.
2. D. D. Anderson and M. Winders, Idealization of a module, Comm. Algebra, (1)1 (2009), 3-56.
3. H. Ansari Toroghi and S. S. Pourmortazavi, On S-primary submodules, Int.Electronic J. Algebra, 31 (2022), 74-89.
4. E. M. Bouba, N. Mahdou and M. Tamekkante, Duplication of a module along an ideal, Acta Math. Hungar, 154 (2018), 29-42.
5. M. D'Anna and M. Fontana, An amalgamated duplication of a ring along an ideal: the basic properties, J. Algebra Appl., 6 (2007), 443-459.
6. S. Ebrahimi Atani and F. Farzalipour, On weakly prime submodules, Tamkang J. of Math., (3) 28 (2007), 247-252.
7. Z. A. El-Bast and P. F. Smith, Multiplication modules, Comm. Algebra, 16 (1988), 755-779.
8. R. El Khalfaoui, N. Mahdou, P. Sahandi and N. Shirmohammadi, Amalgamated modules along an ideal, Comm. Korean Math. Soc., 36 (2021), 1-10.
9. F. Farzalipour, On almost semiprime submodules, Algebra, 2014 (2014), Article ID 752858, 6 pages.
10. F. Farzalipour and P. Ghiasvand, A generalization of graded prime submodules over non-commutative graded rings, J. Algebra Relat. Topics, (1) 8 (2020), 39-50.
11. F. Farzalipour and P. Ghiasvand, On S-1-absorbing prime submodules, J. Algebra Appl., (6) 21 (2022), 2250115, 14 pages.
12. H. A. Khashan, On almost prime submodules, Acta Mathematica Scientia, 32 (2012), 645-651.
13. A. Pekin, U. Tekir and  O. Klc, S-semiprime submodules and S-reduced modules, J. Math., 2020 (2020), Article ID 8824787, 7 pages.
14. S. Rajaee, S-small and S-essential submodules, J. Algebra Relat. Topics, (1) 10 (2022), 1-10.
15. E. S. Sevim, T. Arabaci, U. Tekir and S. Koc, On S-prime submodules, Turk. J. Math., (2) 43 (2019), 1036-1046.
16. G. Ulucak, U. Tekir and S. Koc, On S-2-absorbing submodules and vn-regular modules, An. St. Uni. Ovidius Constanta, (2) 28 (2020), 239-257.
17. F. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Singapore: Springer, 2016.
Journal of Algebra and Related Topics
Volume 11, Issue 1
June 2023
Pages 27-42

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