A GENERALIZED IDEAL BASED-ZERO DIVISOR GRAPHS OF NEAR-RINGS

• Published : 2009.04.30

Abstract

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.

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Cited by

1. On generalized zero divisor graph of a poset vol.161, pp.10-11, 2013, https://doi.org/10.1016/j.dam.2012.12.019
2. On zero-divisors of near-rings of polynomials pp.1727-933X, 2018, https://doi.org/10.2989/16073606.2018.1455070