Theorem of transition

Template:Short description 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 $𝔫$ of B, $𝔫\cap A$ is maximal
2. for each maximal ideal $𝔪$ and $𝔪$-primary ideal $Q$ of $A$, ${\mathrm{length}}_B(B/QB)$ is finite and moreover

   ${\mathrm{length}}_B(B/QB)={\mathrm{length}}_B(B/𝔪B){\mathrm{length}}_A(A/Q).$

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

References

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