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 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 $ab=ba$, we present some equivalent conditions of the reverse order laws for Hirano inverses.