Theoretical Mathematics & Applications

A Generalized ideal based-zero divisor graphs of Noetherian regular -near rings (GIBDNR- -NR)

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  • Abstract

    A near-ring N is called a -Near Ring if it is left simple and N0 is the smallest non-zero ideal of N and a -Near Ring is a non-constant near ring. A Commutative ring N with identity is a Noetherian Regular -Near Ring if it is Semi Prime in which every non-unit is a zero divisor and the Zero ideal is Product of a finite number of principle ideals generated by semi prime elements and N is left simple which has N0 = N, Ne = N. In this paper, we introduce the generalized ideal-based zero divisor graph structure of Noetherian Regular - near-ring N, denoted by I(N). It is shown that if is a completely reflexive ideal of N, then every two vertices in I(N) are connected by a path of length at most 3, and if I(N) contains a cycle, then the core K of I(N) is a union of triangles and rectangles. We have shown that if I(N) is a bipartite graph for a completely semi-prime ideal I of N, then N has two prime ideals whose intersection is I.