The total graph of non-zero annihilating ideals of a commutative ring
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Assume that $R$ is a commutative ring with non-zero identity which is not an integral domain. An ideal $I$ of $R$ is called an annihilating ideal if there exists a non-zero element $a\in R$ such that $Ia=0$. S. Visweswaran and H. D. Patel associated a graph with the set of all non-zero annihilating ideals of $R$, denoted by $\Omega(R)$, as the graph with the vertex-set $\mathrm{A}(R)^*$, the set of all non-zero annihilating ideals of $R$, and two distinct vertices $I$ and $J$ are adjacent if $I+J$ is an annihilating ideal. In this paper, we study the relations between the diameters of $\Omega(R)$ and $\Omega(R[x])$. Also, we study the relations between the diameters of $\Omega(R)$ and $\Omega(R[[x]])$, whenever $R$ is a Noetherian ring. In addition, we investigate the relations between the diameters of this graph and the zero-divisor graph. Moreover, we study some combinatorial properties of $\Omega(R)$ such as domination number and independence number. Furthermore, we study the complement of this graph.
Abstract
Assume that $R$ is a commutative ring with non-zero identity which is not an integral domain. An ideal $I$ of $R$ is called an annihilating ideal if there exists a non-zero element $a\in R$ such that $Ia=0$. S. Visweswaran and H. D. Patel associated a graph with the set of all non-zero annihilating ideals of $R$, denoted by $\Omega(R)$, as the graph with the vertex-set $\mathrm{A}(R)^*$, the set of all non-zero annihilating ideals of $R$, and two distinct vertices $I$ and $J$ are adjacent if $I+J$ is an annihilating ideal. In this paper, we study the relations between the diameters of $\Omega(R)$ and $\Omega(R[x])$. Also, we study the relations between the diameters of $\Omega(R)$ and $\Omega(R[[x]])$, whenever $R$ is a Noetherian ring. In addition, we investigate the relations between the diameters of this graph and the zero-divisor graph. Moreover, we study some combinatorial properties of $\Omega(R)$ such as domination number and independence number. Furthermore, we study the complement of this graph.
참고문헌 (12)
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