Theorem of transition

In algebra, the theorem of transition is said to hold between commutative rings $$A \subset B$$ if
 * 1) $$B$$ dominates $$A$$; i.e., for each proper ideal I of A, $$IB$$ is proper and for each maximal ideal $$\mathfrak n$$ of B, $$\mathfrak n \cap A$$ is maximal
 * 2) for each maximal ideal $$\mathfrak m$$ and $$\mathfrak m$$-primary ideal $$Q$$ of $$A$$, $$\operatorname{length}_B (B/ Q B)$$ is finite and moreover
 * $$\operatorname{length}_B (B/ Q B) = \operatorname{length}_B (B/ \mathfrak{m} B) \operatorname{length}_A(A/Q).$$

Given commutative rings $$A \subset B$$ such that $$B$$ dominates $$A$$ and for each maximal ideal $$\mathfrak m$$ of $$A$$ such that $$\operatorname{length}_B (B/ \mathfrak{m} B)$$ is finite, the natural inclusion $$A \to B$$ is a faithfully flat ring homomorphism if and only if the theorem of transition holds between $$A \subset B$$.