Colouring of Multiplication Modules

Rezvan Varmazyar
Department of Mathematics, Khoy Branch, Islamic Azad University, Khoy, Iran



Content: Let $R$ be a commutative ring. An $R$-module $M$ is called a multiplication $R$-module provided that for every submodule $N$ of $M$ there exists an ideal $I$ of $R$ such that $N=IM$. Let $N=IM$ and $K=JM$ be submodules of a multiplication $R$-module $M$ ,for some ideals $I, J$ of $R$. The product of $N$ and $K$, denoted by $N.K$, is defined to be $(IJ)M$. An element $x$ of $M$ is called a zero-divisor of $M$ if there exists a nonzero element $y$ of $M$ such that $Rx.Ry=0$. We associate a simple graph $\Gamma(M)$ to $M$ with vertices $Z(M)^{\ast}=Z(M)-\{0\}$ ,the set of nonzero zero-divisors of $M$, and for distinct $x, y\in Z(M)^{\ast}$ the vertices $x$ and $y$ are adjacent if and only if $Rx.Ry=0$. In this present talk we investigate the chromatic number of $\Gamma(M)$.

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