LAUSR

Let [Formula: see text] be a semiprime ring, not necessarily with unity, with extended centroid [Formula: see text]. For [Formula: see text], let [Formula: see text] (respectively [Formula: see text], [Formula: see text]) denote the set of all outer (respectively inner, reflexive) inverses of [Formula: see text] in [Formula: see text]. In the paper, we study the inclusion properties of [Formula: see text], [Formula: see text] and [Formula: see text]. Among other results, we prove that for [Formula: see text] with [Formula: see text] von Neumann regular, [Formula: see text] (respectively [Formula: see text]) if and only if [Formula: see text] (respectively [Formula: see text]). Here, [Formula: see text] is the smallest idempotent in [Formula: see text] such that [Formula: see text]. This gives a common generalization of several known results.