On a Generalized Zero-divisor Graph of a Commutative Ring with Respect to an ideal

Priyanka Pratim Baruah, Kuntala Patra

Abstract


In this paper, we generalize the notion of the ideal-based zero-divisor graph of a commutative ring. Let \(R\) be a commutative ring and let \(I\) be an ideal of \(R\). Here, we define a generalized zero-divisor graph of \(R\) with respect to \(I\) and denote this graph by \(\Gamma_I^G(R)\). We show that \(\Gamma_I^G(R)\) is connected with diameter at most three. If \(\Gamma_I^G(R)\) has a cycle, we show that the girth of \(\Gamma_I^G(R)\) is at most four. Also, we investigate the existence of cut vertices of \(\Gamma_I^G(R)\). Moreover,we examine certain situations when \(\Gamma_I^G(R)\) is a complete bipartite graph.

Keywords


Commutative ring, Ideal, Generalized zero-divisor graph, Diameter, Girth.

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References


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