Incompatible transformations of principal ideal rings

We consider problems on transformations of principal ideal rings incompatible with some epimorphisms of a given ring. We introduce the notion of a maximally incompatible transformation and describe all maximally incompatible transformations of the principal ideals ring $A$ by a homomorphism $A\to A\mathfrak m$, where $\mathfrak m$ is some ideal of the ring $A$.