  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.
