Theorem of transition

In algebra, the theorem of transition is said to hold between commutative rings $A\subset B$ if

• (i) $B$ dominates $A$,
• (ii) for each maximal ideal ${\mathfrak {m}}$ and ${\mathfrak {m}}$-primary ideal $Q$ of $A$, $\operatorname {length} _{B}(B/QB)$ is finite and moreover

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

Given rings $A\subset B$ such that $B$ dominates $A$ and for each maximal ideal ${\mathfrak {m}}$ of $A$, $A\to B$ is a faithfully flat ring homomorphism if and only if the theorem of transition hold between $A\subset B$.

References

• Matsumura, Hideyuki (1986). Commutative ring theory. Cambridge Studies in Advanced Mathematics. Vol. 8. Cambridge University Press. ISBN 0-521-36764-6. MR 0879273. Zbl 0603.13001. {{cite book}}: Invalid |ref=harv (help)