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arXiv · 2408.05949

Strong zero-divisor graph of p.q.-Baer $*$-rings

Anil Khairnar·Nana Kumbhar·B. N. Waphare
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Abstract

In this paper, we study the strong zero-divisor graph of a p.q.-Baer $*$-ring. We determine the condition on a p.q.-Baer $*$-ring (in terms of the smallest central projection in a lattice of central projections of a $*$-ring), so that its strong zero-divisor graph contains a cut vertex. It is proved that the set of cut vertices of a strong zero-divisor graph of a p.q.-Baer $*$-ring forms a complete subgraph. We prove that the complement of the strong zero-divisor graph of a p.q.-Baer $*$-ring is connected if and only if the $*$-ring contains at least six central projections. We characterize the diameter and girth of the complement of a strong zero-divisor graph of a p.q.-Baer $*$-ring. Also, we characterize p.q.-Baer $*$-rings whose strong zero-divisor graph is complemented.

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Anil Khairnar, Nana Kumbhar, B. N. Waphare. 2024-08-12. Strong zero-divisor graph of p.q.-Baer $*$-rings. https://arxiv.org/abs/2408.05949

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