Commun. Korean Math. Soc. 2009; 24(2): 161-169
Printed June 1, 2009
https://doi.org/10.4134/CKMS.2009.24.2.161
Copyright © The Korean Mathematical Society.
Patchirajulu Dheena and Balasubramanian Elavarasan
Annamalai University and K. S. R. College of Engineering
In this paper, we introduce the generalized ideal-based zero-divisor graph structure of near-ring $N,$ denoted by $\widehat{\Gamma_{I}(N)}.$ It is shown that if $I$ is a completely reflexive ideal of $N,$ then every two vertices in $\widehat{\Gamma_{I}(N)}$ are connected by a path of length at most 3, and if $\widehat{\Gamma_{I}(N)}$ contains a cycle, then the core $K$ of $\widehat{\Gamma_{I}(N)}$ is a union of triangles and rectangles. We have shown that if $\widehat{\Gamma_{I}(N)}$ is a bipartite graph for a completely semiprime ideal $I$ of $N,$ then $N$ has two prime ideals whose intersection is $I.$
Keywords: ideal-based zero-divisor graph, diameter, near-ring, ideal and cycle
MSC numbers: 16Y30, 13A15
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