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It is the undirected graph with all $M$-regular elements of $R$ as vertices, and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\\in Z(M)$. The basic properties and possible structures of the $M$-$Reg(\\Gamma(R))$ are studied. 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Heydari, M.J. Nikmehr","submitted_at":"2013-05-27T12:45:22Z","abstract_excerpt":"Let $R$ be a commutative ring and $M$ be an $R$-module, and let $Z(M)$ be the set of all zero-divisors on $M$. In 2008, D.F. Anderson and A. Badawi introduced the regular graph of $R$. In this paper, we generalize the regular graph of $R$ to the \\textit{$M$-regular graph} of $R$, denoted by $M$-$Reg(\\Gamma(R))$. It is the undirected graph with all $M$-regular elements of $R$ as vertices, and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\\in Z(M)$. The basic properties and possible structures of the $M$-$Reg(\\Gamma(R))$ are studied. 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