A generalized inverse of an m × n matrix A over a pseudoring means an n × m matrix G satisfying AGA = A. In this paper we give a characterization of matrices having generalized inverses. Also, we introduce and study a space decomposition of a matrix, and prove that a matrix is decomposable if and only if it has a generalized inverse. Finally, we establish necessary and sufficient conditions for a matrix to possess various types of g-inverses including Moore–Penrose inverse.