Theoretical Mathematics & Applications
A Generalized ideal based-zero divisor graphs of Noetherian regular ä-near rings (GIBDNR- ä-NR)
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Abstract
A near-ring N is called a ä-Near – Ring if it is left simple and N0 is the smallest non-zero ideal of N and a ä-Near – Ring is a non-constant near ring. A Commutative ring N with identity is a Noetherian Regular ä-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 ä- near-ring N, denoted by ÃI(N). It is shown that if à is a completely reflexive ideal of N, then every two vertices in ÃI(N) are connected by a path of length at most 3, and if ÃI(N) contains a cycle, then the core K of ÃI(N) is a union of triangles and rectangles. We have shown that if ÃI(N) is a bipartite graph for a completely semi-prime ideal I of N, then N has two prime ideals whose intersection is I.
