Communications of the
Korean Mathematical Society CKMS

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