Theorem of transition

In algebra, the theorem of transition is said to hold between commutative rings if

dominates ; i.e., for each proper ideal I of A, is proper and for each maximal ideal of B, is maximal
for each maximal ideal and -primary ideal of, is finite and moreover
:

Given commutative rings such that dominates and for each maximal ideal of such that is finite, the natural inclusion is a faithfully flat ring homomorphism if and only if the theorem of transition holds between.