Graded generalized 2-absorbing submodules
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Graded generalized 2-absorbing submodules Rashid Abu-Dawwas1 · Malik Bataineh2 · Heba Shashan1 Received: 6 August 2020 / Accepted: 16 October 2020 © The Managing Editors 2020
Abstract Let G be a group, R a G-graded commutative ring with nonzero unity, and M a G-graded R-module. In this article, we introduce the concept of graded generalized 2-absorbing submodules as a generalization of graded 2-absorbing submodules, and investigate some properties of this new class of graded submodules. A proper graded R-submodule N of M is said to be a graded generalized 2-absorbing R-submodule of M if whenever x and y are homogeneous elements of R and m is a homogeneous element of M such that x ym ∈ N , then either it is the case that x or y is in the graded radical of (N : R m), or x y ∈ (N : R M). Keywords Graded 2-absorbing submodules · Graded 2-absorbing Primary submodules · Graded prime submodules Mathematics Subject Classification Primary 16W50; Secondary 13A02
1 Introduction Throughout this paper, all rings are commutative with nonzero unity and all modules are unitary. Furthermore, G will always represent a group. Let M be a graded Rmodule. As Atani (2006), a proper graded R-submodule N of M is said to be graded prime if for any homogeneous elements r ∈ R and m ∈ M such that r m ∈ N , we have either m ∈ N or r ∈ (N : R M). Al-Zoubi et al. (2019) gave a generalization of graded
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Malik Bataineh [email protected] Rashid Abu-Dawwas [email protected] Heba Shashan [email protected]
1
Department of Mathematics, Yarmouk University, Irbid, Jordan
2
Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan
123
Beitr Algebra Geom
prime ideals and called such graded ideals graded 2-absorbing ideals. A proper graded ideal P of R is said to be graded 2-absorbing if whenever a, b, c are homogeneous elements of R such that abc ∈ P, then either ab ∈ P or ac ∈ P or bc ∈ P. In Soheilnia and Darani (2017), Soheilnia and Darani introduced the concept of graded 2-absorbing primary ideals which is a generalization of graded primary ideals. A proper graded ideal P of R is called a graded 2-absorbing primary ideal of R if whenever a, b, c are homogeneous elements of R with abc ∈ P, then ab ∈ P or one of ac or bc is in the graded radical of P. Al-Zoubi and Abu-Dawwas (2014) extended graded 2-absorbing ideals to graded 2-absorbing submodules. A proper graded R-submodule N of M is said to be graded 2-absorbing if whenever a and b are homogeneous elements of R and m is a homogeneous element of M such that abm ∈ N , then either am ∈ N or bm ∈ N or ab ∈ (N : R M). Graded 2-absorbing submodules were deeply studied in Al-Zoubi and Al-Azaizeh (2019). In this article, we follow Farshadifar and Ansari-Toroghy (2020) to introduce the concept of graded generalized 2-absorbing submodules as a generalization of graded 2-absorbing submodules, and investigate some properties of this new class of graded submodules. A proper graded R-submodule N of M is said to be a graded generalized 2-absorbing R
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