Some results on the sum-annihilating essential submodule graph

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Science, University of Payame Noor (PNU), P.O. Box 19395-3697, Tehran, Iran

Abstract

Consider a commutative ring $R$ with a non-zero identity $1\neq 0$, and let $M$ be a non-zero unitary module over $R$. In this document, our goal is to present the sum-annihilating essential submodule graph $\mathbb{AE}^{0}_{R}(M)$ and its subgraph $\mathbb{AE}^{1}_{R}(M)$ of a module $M$ over a commutative ring $R$ which is described in the following way: The vertex set of graph $\mathbb{AE}^{0}_{R}(M)$ (resp., $\mathbb{AE}^{1}_{R}(M)$) is the collection of all (resp., non-zero proper) annihilating submodules of $M$ and two separate annihilating submodules $N$ and $K$ are connected anytime $N+K$ is essential in $M$. We study and investigate the basic properties of graphs $\mathbb{AE}^{i}_{R}(M)$ ($i=0, 1$) and will present some related results. Additionally, we explore how the properties of graphs interact with the algebraic structures they represent.

Keywords

References

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Algebraic Structures and Their Applications

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Available Online from 16 June 2024

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