Strong metric dimension of clean graphs of commutative rings

Let [Formula: see text] be a ring with unity. The clean graph [Formula: see text] of a ring [Formula: see text] is the simple undirected graph whose vertices are of the form [Formula: see text], where [Formula: see text] is an idempotent element and [Formula: see text] is a unit of the ring [Formula: see text], and two vertices [Formula: see text], [Formula: see text] of [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. In this paper, for a commutative ring [Formula: see text], first we obtain the strong resolving graph of [Formula: see text] and its independence number. Using them, we determine the strong metric dimension of the clean graph of an arbitrary commutative ring. As an application, we compute the strong metric dimension of [Formula: see text], where [Formula: see text] is a commutative Artinian ring.