Authors
- E. Vatandoost ^{} ^{1}
- F. Ramezani ^{1}
- A. Bahraini ^{2}
^{1} Faculty of Basic Science, Imam Khomeini International University, Qazvin, Iran
^{2} Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran
Abstract
Let $R$ be a non-commutative ring with unity. The commuting graph of $R$ denoted by $\Gamma(R)$, is a graph with vertex set $R\Z(R)$ and two vertices $a$ and $b$ are adjacent iff $ab=ba$. In this paper, we consider the commuting graph of non-commutative rings of order pq and $p^2q$ with Z(R) = 0 and non-commutative rings with unity of order $p^3q$. It is proved that $C_R(a)$ is a commutative ring for every $0\neq a \in R\Z(R)$. Also it is shown that if $a,b\in R\Z(R)$ and $ab\neq ba$, then $C_R(a)\cap C_R(b)= Z(R)$. We show that the commuting graph $\Gamma(R)$ is the disjoint union of $k$ copies of the complete graph and so is not a connected graph.
Keywords
Main Subjects
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