Some results on the reverse order law in rings with involution

Abstract

We investigate some necessary and sufficient conditions for the hybrid reverse order law (ab) # = b † a † in rings with involution. Assuming that a and b are Moore-Penrose invertible, we present an equivalent condition for the product ab to be an EP element.Mathematics Subject Classification (2010). 16B99, 15A09, 46L05.

On forward-order law for core inverse in rings

“…Zhu ([21] and [22]) conferred several results on additive properties, reverse-order law and forward-order law. In 2012, Mosic and Djordjevic [13] provided the hybrid reverse-order law between the group inverse and the Moore-Penrose inverse. In 2018, Zhu et al [20] provided the reverse-order law for the generalized core inverse.…”

Reverse Order Law for the Core Inverse in Rings

“…Recently, Mary [19] provided equivalent conditions for the two-sided reverse order law (ab) # = b # a # and (ba) # = a # b # for the group inverse in semigroups and rings. In [20,22,24], Mosić et al considered the mixed-type reverse order laws in rings, such as (ab) † = b † (a † abb † ) † a † , (ab) # = b † (a † abb † ) † a † , (ab) # = (a † ab) † a † , and (ab) # = b † a † . More results on the reverse order law for the generalized inverse can be found in [3, 7-9, 11, 16-18, 21, 25, 29].…”

“…The article is motivated by the papers [19,22,30]. We present some equivalent conditions for the one-sided reverse order law (ab) # = b # a # , the two-sided reverse order law (ab) # = b # a # and (ba) # = a # b # for the core inverse in rings with involution.…”

The diamond partial order in rings

“…Taking x − = x # we get xx − = x 2 = yx = yx − and x − x = x − y is similar. A wide range of properties related to these orders and the generalized inverses involved in each of them can be found in [1,2,3,6,7,11,12,13]. This paper is organized as follows.…”