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This article concerns the MP inverse of the differences and the products ofprojections in a ring $R$ with involution. Some equivalent conditions areobtained. As applications, the MP invertibility of the commutator $pq-qp$ andthe anti-commutator $pq+qp$ are characterized, where $p$ and $q$ areprojections in $R$. Some related known results in $C^*$-algebra aregeneralized.

Moore-Penrose Invertibility of Differences and Products of Projections in Rings with Involution