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.
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