A GRAPH ASSOCIATED TO ESPECIAL ESSENTIALITY OF SUBMODULES

‎Let $R$ be an associative ring with identity‎. ‎In this paper we‎‎associate 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 distinct‎‎vertices $K$ and $L$ are considered to be adjacent if and only‎‎if $K\cap L$ is an essential submodule of $K+L$‎.‎‎We investigate the relationship between some module theoretic‎‎properties of $M$ such as minimality and closedness of‎‎submodules with some graph theoretic properties of‎‎$\Gamma_e(M)$‎. ‎In general‎, ‎this graph is not connected‎. ‎We‎‎study some special cases in which $\Gamma_e(M)$ is‎‎complete or a union of complete connected components and give some examples illustrating each specific case‎.