The relationship between the Bezout and input-output approaches to rig
ht-coprimeness is investigated. We introduce a generalized Bezout iden
tity and analyze its relationship to right-coprimeness. In terms of th
e generalized Bezout identity, a right factorization of a nonlinear pl
ant 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.