theorem of transition

{{short description|Theorem about commutative rings and subrings}}

In algebra, the theorem of transition is said to hold between commutative rings $A\; \backslash subset\; B$ if{{harvnb|Nagata|1975|loc=Ch. II, § 19.}}{{harvnb|Matsumura|1986|loc=Ch. 8, Exercise 22.1.}}

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

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