Commun. Korean Math. Soc. 2019; 34(2): 415-427
Online first article November 20, 2018 Printed April 30, 2019
https://doi.org/10.4134/CKMS.c180193
Copyright © The Korean Mathematical Society.
Abolfazl Alibemani, Ebrahim Hashemi
Shahrood University of Technology; Shahrood University of Technology
Let $R$ be an associative ring with nonzero identity. The annihilator ideal graph of $R$, denoted by $\Gamma_{\mathrm{Ann}}(R)$, is a graph whose vertices are all nonzero proper left ideals and all nonzero proper right ideals of $R$, and two distinct vertices $I$ and $J$ are adjacent if $I\cap (\ell_R(J)\cup r_R(J))\neq0$ or $J\cap (\ell_R(I)\cup r_R(I))\neq0$, where $\ell_R(K)=\{b\in R~|~bK=0\}$ is the left annihilator of a nonempty subset $K\subseteq R$, and $r_R(K)=\{b\in R~|~Kb=0\}$ is the right annihilator of a nonempty subset $K\subseteq R$. In this paper, we assume that $R$ is a semicommutative ring. We study the structure of $\Gamma_{\mathrm{Ann}}(R)$. Also, we investigate the relations between the ring-theoretic properties of $R$ and graph-theoretic properties of $\Gamma_{\mathrm{Ann}}(R)$. Moreover, some combinatorial properties of $\Gamma_{\mathrm{Ann}}(R)$, such as domination number and clique number, are studied.
Keywords: domination number, annihilator ideal graph, reversible ring, semicommutative ring
MSC numbers: 16U99, 05C69
2021; 36(2): 209-227
2017; 32(4): 811-826
2012; 27(4): 843-849
2010; 25(3): 349-364
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd