The theory of factorization of rational \(n\times n\) matrix functions \(W(\lambda)=D+C(\lambda -A)^{-1}B,\) as developed by H. Bart, I. Gohberg and M. A. Kaashoek [Minimal factorization of matrix and operator functions (1979; Zbl 0424.47001)], is extended to the case where \(D=W(\infty)\) is not invertible. This extension requires a generalization of the notion of a supporting projection and includes a description of all minimal factorizations of which the factors are again square rational matrix functions. Special attention is paid to spectral factorizations, stable factorizations and to the case when \(\det W(\lambda)\not\equiv {\text{ß}}.\) As an application a new proof is given of the Gohberg- Lancaster-Rodman factorization theorem for monic matrix polynomials.