A GENERALIZATION OF THE ZERO-DIVISOR GRAPH FOR MODULES

Abstract

Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say <TEX>${\Gamma}(M)$</TEX>, such that when M = R, <TEX>${\Gamma}(M)$</TEX> is exactly the classic zero-divisor graph. Many well-known results by D. F. Anderson and P. S. Livingston, in [5], and by D. F. Anderson and S. B. Mulay, in [6], have been generalized for <TEX>${\Gamma}(M)$</TEX> in the present article. We show that <TEX>${\Gamma}(M)$</TEX> is connected with <TEX>$diam({\Gamma}(M)){\leq}3$</TEX>. We also show that for a reduced module M with <TEX>$Z(M)^*{\neq}M{\backslash}\{0\}$</TEX>, <TEX>$gr({\Gamma}(M))={\infty}$</TEX> if and only if <TEX>${\Gamma}(M)$</TEX> is a star graph. Furthermore, we show that for a finitely generated semisimple R-module M such that its homogeneous components are simple, <TEX>$x,y{\in}M{\backslash}\{0\}$</TEX> are adjacent if and only if <TEX>$xR{\cap}yR=(0)$</TEX>. Among other things, it is also observed that <TEX>${\Gamma}(M)={\emptyset}$</TEX> if and only if M is uniform, ann(M) is a radical ideal, and <TEX>$Z(M)^*{\neq}M{\backslash}\{0\}$</TEX>, if and only if ann(M) is prime and <TEX>$Z(M)^*{\neq}M{\backslash}\{0\}$</TEX>.

Locations

• Journal of the Korean Mathematical Society - View - PDF