In proper ∗-rings, we characterize weak group inverses by three equations. It generalizes the notion of weak group inverse, which was introduced by Wang and Chen for complex matrices in 2018. Some new equivalent characterizations for elements to be weak group invertible are presented. Furthermore, we define the group-EP decomposition. Some properties of the weak group inverse are established by the group-EP decomposition.

No abstract

A weak group element is introduced in a proper * -ring. Several equivalent conditions of weak group elements are investigated. We prove that an element is pseudo core invertible if it is both partial isometry and weak group invertible. Reverse order law and additive property of the weak group inverse are presented. Finally, under certain assumption on a, equivalent conditions of a W a * = a * a W are presented by using the normality of the group invertible part of an element in its group-EP decomposition.

No abstract

Let R be a unital ring with involution. The pseudo core inverses of differences and products of two projections are investigated in R. Some equivalent conditions are obtained. As applications, the pseudo core invertibility of the commutator pq - qp and the anti-commutator pq + qp are characterized, where p; q are projections in R.