On the metric dimension of a total graph of non-zero annihilating ideals

Abstract

Abstract Let R be a commutative ring with identity which is not an integral domain. An ideal I of a ring R is called an annihilating ideal if there exists r ∈ R − {0} such that Ir = (0). Visweswaran and H. D. Patel associated a graph with the set of all non-zero annihilating ideals of R, denoted by Ω(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, J are joined if and only if I +J is also an annihilating ideal of R. In this paper, we study the metric dimension of Ω(R) and provide metric dimension formulas for Ω(R).