Vatandoost, E., Ramezani, F., Bahraini, A. (2014). On the commuting graph of non-commutative rings of order $p^nq$. Journal of Linear and Topological Algebra (JLTA), 03(01), 1-6.

E. Vatandoost; F. Ramezani; A. Bahraini. "On the commuting graph of non-commutative rings of order $p^nq$". Journal of Linear and Topological Algebra (JLTA), 03, 01, 2014, 1-6.

Vatandoost, E., Ramezani, F., Bahraini, A. (2014). 'On the commuting graph of non-commutative rings of order $p^nq$', Journal of Linear and Topological Algebra (JLTA), 03(01), pp. 1-6.

Vatandoost, E., Ramezani, F., Bahraini, A. On the commuting graph of non-commutative rings of order $p^nq$. Journal of Linear and Topological Algebra (JLTA), 2014; 03(01): 1-6.

On the commuting graph of non-commutative rings of order $p^nq$

^{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.

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