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In this article, we give several generalizations of the concept of zero-divisor elements
in a commutative ring with identity to modules. Then for each $R$-module
$M$ we associate three undirected (simple) graphs $\Gamma^*(_RM)\subseteq \Gamma(_RM)\subseteq \Gamma_*(_RM)$
which, for $M=R$ all coincide with the zero-divisor graph of $R$. The main object
of this paper is to study the interplay of module-theoretic properties of $M$ with
graph-theoretic properties of these graphs.
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