Hua’s identity and super Jacobson’s lemma for generalized inverses in rings

Given an associative ring R and a,b∈R, a known result of Brown and McCoy states that the (von Neumann) regularity of a−aba implies that of a (but not conversely). Inspired by a modification of Hua’s identity a−aba=[a−1+(b−1−a)−1]−1, we show that a−aba is regular if and only if both a and 1−ab are regular, and present a generalization of Hua’s identity involving inner inverses of a−aba, a and 1−ab. Moreover, for any a,b,x∈R, we refine Super Jacobson’s lemma in the sense of Khurana and Lam by giving the connection between inverses or generalized inverses of the two elements a+b−axb and a+b−bxa in a Super Jacobson pair. Along a related direction, we also give a stronger version of Hua’s identity.