Let R be a ring with unity and let a, ∈ R be such that a is regular. In this article, the generalized invertibility of a + 1 − aa − are investigated in term of the generalized invertibility of elements in a corner ring. As applications, several equivalent conditions on the Drazin invertibility of product and difference of idempotents are obtained. Moreover, we present the equivalent conditions for the existence of Moore-Penrose inverse in a ring with involution.