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

This article concerns the MP inverse of the differences and the products of projections in a ring R   with involution. Some equivalent conditions are obtained. As applications, the MP invertibility of the commutator pq−qppq−qp and the anti-commutator pq+qppq+qp are characterized, where p and q are projections in R  . Some related known results in C⁎C⁎-algebra are generalized.