DOI QR코드

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A GENERALIZATION OF THE ZERO-DIVISOR GRAPH FOR MODULES

  • 투고 : 2012.10.03
  • 발행 : 2014.01.01

초록

Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say ${\Gamma}(M)$, such that when M = R, ${\Gamma}(M)$ 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 ${\Gamma}(M)$ in the present article. We show that ${\Gamma}(M)$ is connected with $diam({\Gamma}(M)){\leq}3$. We also show that for a reduced module M with $Z(M)^*{\neq}M{\backslash}\{0\}$, $gr({\Gamma}(M))={\infty}$ if and only if ${\Gamma}(M)$ is a star graph. Furthermore, we show that for a finitely generated semisimple R-module M such that its homogeneous components are simple, $x,y{\in}M{\backslash}\{0\}$ are adjacent if and only if $xR{\cap}yR=(0)$. Among other things, it is also observed that ${\Gamma}(M)={\emptyset}$ if and only if M is uniform, ann(M) is a radical ideal, and $Z(M)^*{\neq}M{\backslash}\{0\}$, if and only if ann(M) is prime and $Z(M)^*{\neq}M{\backslash}\{0\}$.

키워드

참고문헌

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피인용 문헌

  1. A conception of zero-divisor graph for categories of modules vol.15, pp.01, 2016, https://doi.org/10.1142/S0219498816500122
  2. Zero-divisor graphs for modules over integral domains vol.16, pp.05, 2017, https://doi.org/10.1142/S0219498817500876
  3. ON GRAPHS ASSOCIATED WITH MODULES OVER COMMUTATIVE RINGS vol.53, pp.5, 2016, https://doi.org/10.4134/JKMS.j150457