STUDY OF THE ANNIHILATOR IDEAL GRAPH OF A SEMICOMMUTATIVE RING

摘要

Let R be an associative ring with nonzero identity. The an-nihilator ideal graph of R, denoted by ??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 (`R(J) [ rR(J)) 6= 0 or J (`R(I) [ rR(I)) 6= 0, where `R(K) = fb 2 R j bK = 0g is the left annihilator of a nonempty subset K R, and rR(K) = fb 2 R j Kb = 0g is the right annihilator of a nonempty subset K R. In this paper, we assume that R is a semicommutative ring. We study the structure of ??Ann(R). Also, we investigate the relations between the ring-theoretic properties of R and graph-theoretic properties of ??Ann(R). Moreover, some combinatorial properties of ??Ann(R), such as domination number and clique number, are studied.