Reverse order laws for Hirano inverses in rings

Abstract

In this paper, various kinds of reverse order laws for the Hirano inverse are characterized in a ring with the unit 1. Under some of conditions aba = a 2 b, ab 2 = bab, a h ab 2 = ba h ab, a 2 bb h = abb h a instead of the condition ab = ba, respectively, we present some equivalent conditions of the reverse order laws for Hirano inverses. Definition 1.1. Let a ∈ R. If there exists b ∈ R such that (2) bab = b, (5) ab = ba, (7) a 2 − ab ∈ N(R), then b is called the Hirano inverse of a, and a is Hirano invertible. Remark 1. For a ∈ R, the set of elements satisfying the (i)th equation in Definition 1.1 is denoted by a{i}, where i ∈ {2, 5, 7}. And the set of Hirano invertible elements in R is denoted by R h. Definition 1.2. Let a ∈ R. If there exists b ∈ R such that (2) bab = b, (5) ab = ba, (7) a − a 2 b ∈ N(R), then b is called the Drazin inverse of a, and a is Drazin invertible.