THE TOTAL GRAPH OF NON-ZERO ANNIHILATING IDEALS OF A COMMUTATIVE RING

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 is an element of 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 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.