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 $pq-qp$ and the anti-commutator $pq+qp$ are characterized, where $p$ and $q$ are projections in $R$. Some related known results in $C^*$-algebra are generalized.