On metric dimension and crosscap analysis of sum-annihilating essential ideal graph of commutative rings

Consider a commutative ring with unity denoted as [Formula: see text]. An ideal I of a ring [Formula: see text] is called an annihilating ideal if there exists a non-zero element [Formula: see text] such that rI = 0. An ideal J of [Formula: see text] is called an essential ideal if J has a non-zero intersection with every other non-zero ideal of [Formula: see text]. The sum-annihilating essential ideal graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertex set is the set of all non-zero annihilating ideals of [Formula: see text] and two distinct vertices I and J are adjacent whenever [Formula: see text] is an essential ideal of [Formula: see text]. In this research paper, we have determined the metric dimension of [Formula: see text] for various classifications of rings. Furthermore, we have classified Artinian rings [Formula: see text] for which the sum-annihilating essential ideal graph exhibits projective properties.