Equalities of ideals associated with two projections in rings with involution

Abstract

In this paper we study various right ideals associated with two projections (self-adjoint idempotents) in a ring with involution. Results of O.M. Baksalary, G. Trenkler, R. Piziak, P.L. Odell, and R. Hahn about orthogonal projectors (complex matrices which are Hermitian and idempotent) are considered in the setting of rings with involution. New proofs based on algebraic arguments; rather than finite-dimensional and rank theory; are given.

Projections and Idempotents in star-reducing Rings Involving the Moore-Penrose Inverse

“…Let p and q be two projections in R such that p(1 − q), (1 − p)q ∈ R † , then Proof. Since p(1 − q) ∈ R † , it follows that pR ∩ qR is generated by the projection p − p(pq) † by Lemma 5 (3). Similarly, pR ∩ qR is generated by the projection p − p(pq) † since (1 − p)q ∈ R † .…”

Moore–Penrose invertibility of differences and products of projections in rings with involution

“…The next lemma is essentially due to Benitez and Cvetkovic-Ilic [1] although we do not assume that R is * -reducing. (2) If pq ∈ R † , then bdd † = b;…”