Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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STUDY OF THE ANNIHILATOR IDEAL GRAPH OF A SEMICOMMUTATIVE RING

  • Alibemani, Abolfazl (Faculty of Mathematical Sciences Shahrood University of Technology) ;
  • Hashemi, Ebrahim (Faculty of Mathematical Sciences Shahrood University of Technology)
  • Received : 2018.05.04
  • Accepted : 2018.07.24
  • Published : 2019.04.30
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Abstract

Let R be an associative ring with nonzero identity. The annihilator ideal graph of R, denoted by ${\Gamma}_{Ann}(R)$, 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 $I{\cap}({\ell}_R(J){\cup}r_R(J)){\neq}0$ or $J{\cap}({\ell}_R(I){\cup}r_R(I)){\neq}0$, 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 ${\Gamma}_{Ann}(R)$. Also, we investigate the relations between the ring-theoretic properties of R and graph-theoretic properties of ${\Gamma}_{Ann}(R)$. Moreover, some combinatorial properties of ${\Gamma}_{Ann}(R)$, such as domination number and clique number, are studied.

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References

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