Some results on graded $S$-strongly prime submodules
Subject Areas : Commutative algebra
F.
Farzalipour
1
(دانشگاه پیام نور، دانشکده علوم پایه، گروه ریاضی، تهران، ایران)
Keywords: Graded S-prime submodule, graded S-strongly prime submodule, graded multiplication module,
Abstract :
Let $G$ be a group with identity $e$ and $R$ be a commutative $G$-graded ring with nonzero identity, $S\subseteq h(R)$ a multiplicatively closed subset of $R$ and $M$ be a graded $R$-module. A graded submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ is said to be graded $S$-strongly prime if there exists $s\in S$ such that whenever $((N+Rx_{g}):_{R}M)y_{h}\subseteq N$, then $sx_{g}\in N$ or $sy_{h}\in N$ for all $x_{g},y_{h}\in h(M)$. The aim of this paper is to introduce and investigate some basic properties of the notion of graded $S$-strongly prime submodules, especially in graded multiplication modules. Moreover, we investigate the behaviour of this structure under graded module homomorphisms, localizations of graded modules, quotient graded modules, Cartesian product.
[1] R. Abu-Dawwas, M. Bataineh, Graded prime submodules over non-commutative rings, Vietnam J. Math. 46 (3) (2018), 681-692.
[2] S. Ebrahimi Atani, F. Farzalipour, On graded secondary modules, Turkish J. Math. 31 (2007), 371-378.
[3] J. Escoriza, B. Torrecillas, Multiplication objects in commutative grothendieck categories, Comm. Algebra. 26 (6) (1998), 1867-1883.
[4] F. Farzalipour, P. Ghiasvand, A generalization of graded prime submodules over non-commutative graded rings, J. Algebra. Related Topics. 8 (1) (2020), 39-50.
[5] F. Farzalipour, P. Ghiasvand, Graded S-1-absorbing prime submodules in graded multiplication modules, Int. Electronic J. Algebra. 32 (2022), 62-79.
[6] F. Farzalipour, P. Ghiasvand, On graded weak multiplication modules, Tamkang J. Math. 43 (2) (2012), 171-177.
[7] F. Farzalipour, P. Ghiasvand, On the union of graded prime submodules, Thai. J. Math. 9 (1) (2011), 49-55.
[8] P. Ghiasvand, F. Farzalipour, Generalization of graded second submodules, Acta Universitatis Sapientiae, Mathematica. 13 (1) (2021), 164-181.
[9] P. Ghiasvand, F. Farzalipour, Graded semiprime submodules over non-commutative graded rings, J. Algebraic Sys. 10 (1) (2022), 95-110.
[10] K. Hakan Oral,Ü. Tekir, A. G. Aˇ gargün, On graded prime and primary submodules, Turkish J. Math. 35 (2011), 159-167.
[11] R. McCasland, M. Moore, On radicals of submodules of finitely generated modules, Canad. Math. Bull. 29 (1) (1986), 36-39.
[12] N. Nastasescu, F. Van Oystaeyen, Graded Rings Theory, Mathematical Library 28, North Holland, Amsterdam, 1982.
[13] N. Nastasescu, F. Van Oystaeyen, Methods of Graded Rings, Lecture Notes in Mathematics, Springer, Berlin, 2004.
[14] M. Refai, K. Al-Zoubi, On graded primary ideals, Turkish J. Math. 28 (2004), 217-229.
[15] H. Saber, T. Alraqad, R. Abu-Dawwas, On graded S-prime submodules, AIMS Math. 6 (3) (2020), 2510-2524.