Theorem of transition

In algebra, the theorem of transition is said to hold between commutative rings ${\displaystyle A\subset B}$ if¹²

1. ${\displaystyle B}$ dominates ${\displaystyle A}$; i.e., for each proper ideal I of A, ${\displaystyle IB}$ is proper and for each maximal ideal ${\displaystyle {\mathfrak {n}}}$ of B, ${\displaystyle {\mathfrak {n}}\cap A}$ is maximal
2. for each maximal ideal ${\displaystyle {\mathfrak {m}}}$ and ${\displaystyle {\mathfrak {m}}}$-primary ideal ${\displaystyle Q}$ of ${\displaystyle A}$, ${\displaystyle \operatorname {length} _{B}(B/QB)}$ is finite and moreover

   ${\displaystyle \operatorname {length} _{B}(B/QB)=\operatorname {length} _{B}(B/{\mathfrak {m}}B)\operatorname {length} _{A}(A/Q).}$

Given commutative rings ${\displaystyle A\subset B}$ such that ${\displaystyle B}$ dominates ${\displaystyle A}$ and for each maximal ideal ${\displaystyle {\mathfrak {m}}}$ of ${\displaystyle A}$ such that ${\displaystyle \operatorname {length} _{B}(B/{\mathfrak {m}}B)}$ is finite, the natural inclusion ${\displaystyle A\to B}$ is a faithfully flat ring homomorphism if and only if the theorem of transition holds between ${\displaystyle A\subset B}$.²