Document Type : Original Manuscript


Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, P.O. Box 65719-95863, Malayer, Iran.


‎Let $R$ be an associative ring with identity‎. ‎In this paper we‎
‎associate to every $R$-module $M$ a simple graph $\Gamma_e(M)$‎, which we call it the essentiality graph of $M$. The vertices of $\Gamma_e(M)$ are nonzero submodules of $M$ and two distinct‎
‎vertices $K$ and $L$ are considered to be adjacent if and only‎
‎if $K\cap L$ is an essential submodule of $K+L$‎.

‎We investigate the relationship between some module theoretic‎
‎properties of $M$ such as minimality and closedness of‎
‎submodules with some graph theoretic properties of‎
‎$\Gamma_e(M)$‎. ‎In general‎, ‎this graph is not connected‎. ‎We‎
‎study some special cases in which $\Gamma_e(M)$ is‎
‎complete or a union of complete connected components and give some examples illustrating each specific case‎.


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