the sum-annihilating essential ideal graph of a commutative ring

Let r be a commutative ring with identity. an ideal i of a ring r is called an annihilating ideal if there exists r∈r∖{0} such that ir=(0) and an ideal i of r is called an essential ideal if i has non-zero intersection with every other non-zero ideal of r. the sum-annihilating essential ideal graph of r, denoted by aer, is a graph whose vertex set is the set of all non-zero annihilating ideals and two vertices i and j are adjacent whenever ann(i)+ann(j) is an essential ideal. in this paper we initiate the study of the sum-annihilating essential ideal graph. we first characterize all rings whose sum-annihilating essential ideal graph are stars or complete graphs and then establish sharp bounds on domination number of this graph. furthermore determine all isomorphism classes of artinian rings whose sum-annihilating essential ideal graph has genus zero or one.