Some Remarks on ϕ-Graded Semi-n-Absorbing Submodules

Document Type : Original Research Article

Authors

1 Department of Mathematics, Roudehen Branch, Islamic Azad University, Roudehen, Iran.

2 Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences Semnan University, Semnan, Iran.

Abstract

‎Let $R$ be a $G$-graded commutative ring with identity and $M$ be a unitary $G$-graded $R$-module‎. ‎Let $S(M)$ be the set of all graded submodules of $M$ and $\phi:S(M)\rightarrow S(M)\cup \lbrace\emptyset\rbrace$ be a function‎. ‎A proper graded submodule $N$ of $M$ is called $\phi$-graded semi-$n$-absorbing submodule if whenever $r \in h(R)$ and $m\in h(M)$ with $r^{n}m\in N\backslash\phi(N)$‎, ‎then $r^{n}\in (N:M)$ or $r^{n-1}m\in N$ ($n\geq2$)‎. ‎In this work‎, ‎firstly‎, ‎we state with deeper results on the structure of generalizations of prime submodules as $\phi $-graded prime submodules‎. ‎Moreover‎ $\phi$-graded semi-$n$-absorbing submodules are studied and some results are obtained‎.

Keywords

References

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Analytical and Numerical Solutions for Nonlinear Equations
Volume 8, Issue 2
December 2023
Pages 139-151

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History

  • Receive Date: 10 June 2024
  • Revise Date: 09 December 2024
  • Accept Date: 15 January 2025
  • Publish Date: 01 December 2024

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