The reverse order law of the $(b, c)$-inverse in rings

Abstract

We present equivalent conditions of reverse order law for the $(b, c)$-inverse $(aw)^{(b,c)}=w^{(b,s)}a^{(t,c)}$ to hold in a ring. Also, we study various mixed-type reverse order laws for the $(b,c)$-inverse. As a consequence, we get results related to the reverse order law for the inverse along an element. More general case of reverse order law $(a_1a_2)^{(b_3, c_3)}=a_2^{(b_2, c_2)}a_1^{(b_1, c_1)}$ is considered too.