Nilpotent graphs of skew polynomial rings over non-commutative rings
Publish place: Transactions on Combinatorics، Vol: 9، Issue: 1
Publish Year: 1399
نوع سند: مقاله ژورنالی
زبان: English
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شناسه ملی سند علمی:
JR_COMB-9-1_004
تاریخ نمایه سازی: 14 اردیبهشت 1400
Abstract:
Let $R$ be a ring and $\alpha$ be a ring endomorphism of $R$. The undirected nilpotent graph of $R$, denoted by $\Gamma_N(R)$, is a graph with vertex set $Z_N(R)^*$, and two distinct vertices $x$ and $y$ are connected by an edge if and only if $xy$ is nilpotent, where $Z_N(R)=\{x\in R\;|\; xy\; \rm{is\; nilpotent,\;for\; some}\; y\in R^*\}.$ In this article, we investigate the interplay between the ring theoretical properties of a skew polynomial ring $R[x;\alpha]$ and the graph-theoretical properties of its nilpotent graph $\Gamma_N(R[x;\alpha])$. It is shown that if $R$ is a symmetric and $\alpha$-compatible with exactly two minimal primes, then $diam(\Gamma_N(R[x,\alpha]))=۲$. Also we prove that $\Gamma_N(R)$ is a complete graph if and only if $R$ is isomorphic to $\Z_۲\times\Z_۲$.
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Authors
Mohammad Javad Nikmehr
K.N.Toosi University
Abdolreza Azadi
K. N. Toosi University of Technology