Study of the annihilator ideal graph of a semicommutative ring
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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.
Abstract
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.
참고문헌 (14)
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