Graded S-1-absorbing prime submodules in graded multiplication modules

Year 2022, , 62 - 79, 16.07.2022

Farkhondeh Farzalıpour Peyman Ghıasvand

https://doi.org/10.24330/ieja.1081701

Cited By: 1

Abstract

Let $G$ be a group with identity $e$. Let $R$ be a commutative $G$-graded ring with non-zero identity, $S\subseteq h(R)$ a multiplicatively closed subset of $R$ and $M$ a graded $R$-module. In this article, we introduce and study the concept of graded $S$-1-absorbing prime submodules. A graded submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ is said to be graded $S$-1-absorbing prime, if there exists an $s_{g}\in S$ such that whenever $a_{h}b_{h'}m_{k}\in N$, then either $s_{g}a_{h}b_{h'}\in (N:_{R}M)$ or $s_{g}m_{k}\in N$ for all non-unit elements $a_{h},b_{h'}\in h(R)$ and all $m_{k}\in h(M)$. Some examples, characterizations and properties of graded $S$-1-absorbing prime submodules are given. Moreover, we give some characterizations of graded $S$-1-absorbing prime submodules in graded multiplicative modules.

Keywords

Graded S-1-absorbing prime ideal, graded S-1-absorbing prime submodule, graded multiplication module

References

• R. Abu-Dawwas, M. Bataineh and M. Al-Muanger, Graded prime submodules over non-commutative rings, Vietnam J. Math., 46(3) (2018), 681-692.
• K. Al-Zoubi and R. Abu-Dawwas, On graded 2-absorbing and weakly graded 2-absorbing submodules, Journal of Mathematical Sciences: Advances and Applications, 28 (2014), 45-60.
• S. Ebrahimi Atani and F. Farzalipour, Notes on the graded prime submodules, Int. Math. Forum, (1) (2006), 1871-1880.
• S. Ebrahimi Atani and F. Farzalipour, On graded secondary modules, Turkish J. Math., 31 (2007), 371-378.
• J. Escoriza and B. Torrecillas, Multiplication objects in commutative Grothendieck categories, Comm. Algebra, 26(6) (1998), 1867-1883.
• F. Farzalipour and P. Ghiasvand, On $S$-1-absorbing prime submodules, J. Algebra Appl., to appear.
• F. Farzalipour and P. Ghiasvand, On the union of graded prime submodules, Thai. J. Math., 9(1) (2011), 49-55.
• F. Farzalipour and P. Ghiasvand, On graded weak multiplication modules, Tamkang J. Math., 43(2) (2012), 171-177.
• F. Farzalipour, P. Ghiasvand and M. Adlifard, On graded weakly semiprime submodules, Thai. J. Math., 12(1) (2014), 167-174.
• P. Ghiasvand and F. Farzalipour, Some properties of graded multiplication modules, Far East J. Math. Sci., 34(3) (2009), 341-352.
• P. Ghiasvand and F. Farzalipour, On the graded primary radical of graded submodules, Adv. Appl. Math. Sci., 10(1) (2011), 1-7.
• K. H. Oral, Ü. Tekir and A. G. Ağargün, On graded prime and primary submodules, Turkish J. Math., 35 (2011), 159-167.
• N. Nastasescu and F. van Oystaeyen, Graded Ring Theory, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-New York, 1982.
• N. Nastasescu and F. van Oystaeyen, Methods of Graded Rings, Lecture Notes in Mathematics, vol. 1836., Springer-Verlag, Berlin, 2004.