Range projections and the Moore-Penrose inverse in rings with involution

Abstract

In this article we study the existence of range projections in rings with involution, relating it to the existence of the Moore-Penrose inverse. The results are applied to the solution of the equation xbx = x in rings with involution, extending the results of Greville for matrices. Simpler new proofs are given of the Moore-Penrose invertibility of regular elements in rings with involution, and of the Ljance's formula. © 2007 Taylor & Francis.