The group inverse of 2 × 2 matrices over a ring

Firstly, we give conditions for the existence of the group inverse of the 2 × 2 matrixover an arbitrary ring R with unity 1. Our second purpose is to derive a representation for the group inverse of M in the case when either the entry a or d has group inverse in R. We illustrate our results with some numerical examples in the setting of 2 × 2 complex block matrices.