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

On the commuting graph of non-commutative rings of order pnq

^{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 (R), is a graph with vertex set RnZ(R) and two vertices a and b are adjacent i ab = ba.
In this paper, we consider the commuting graph of non-commutative rings of order pq and p2q
with Z(R) = 0 and non-commutative rings with unity of order p3q. It is proved that CR(a)
is a commutative ring for every 0 ̸= a 2 R n Z(R). Also it is shown that if a; b 2 R n Z(R)
and ab ̸= ba, then CR(a) \ CR(b) = Z(R). We show that the commuting graph (R) is the
disjoint union of k copies of the complete graph and so is not a connected graph.