Theorem Of Transition

In algebra, the theorem of transition is said to hold between commutative rings $A \subset B$ if
# $B$ 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
# for each maximal ideal $\mathfrak m$ and $\mathfrak m$-primary ideal $Q$ of $A$, $\operatorname_B (B/ Q B)$ is finite and moreover
#:$\operatorname_B (B/ Q B) = \operatorname_B (B/ \mathfrak B) \operatorname_A(A/Q).$

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

Notes

References

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Theorems in ring theory

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