The relationship between the Bezout and input-output approaches to right-coprimeness is investigated. We introduce a generalized Bezout identity and analyze its relationship to right-coprimeness. In terms of the generalized Bezout identity, a right factorization of a nonlinear plant is coprime according to the input-output definition if and only if it satisfies the generalized Bezout identity. Therefore the stability of an observer-controller configuration is a necessary and sufficient condition for a nonlinear plant to have a right coprime factorization. Finally we present generalized Bezout maps for some nonlinear plants with a state-space representation.