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Nilpotent graphs of skew polynomial rings over non-commutative rings | ||
| Transactions on Combinatorics | ||
| مقاله 21، دوره 9، شماره 1، خرداد 2020، صفحه 41-48 اصل مقاله (229.98 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2019.117529.1651 | ||
| نویسندگان | ||
| Mohammad Javad Nikmehr* 1؛ Abdolreza Azadi2 | ||
| 1K.N.Toosi University | ||
| 2K. N. Toosi University of Technology | ||
| چکیده | ||
| 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 $diam(\Gamma_N(R[x,\alpha]))=2$. Also we prove that $\Gamma_N(R)$ is a complete graph if and only if $R$ is isomorphic to $𝕫_2\times𝕫_2$. | ||
| کلیدواژه‌ها | ||
| Nilpotent graph؛ $alpha$-compatible rings؛ skew polynomial ring؛ symmetric ring؛ diameter, | ||
| مراجع | ||
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