Theorem of transition

Short description: Theorem about commutative rings and subrings

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

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

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

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

References

• Nagata, Local Rings
• Matsumura, Hideyuki (1986). Commutative ring theory. Cambridge Studies in Advanced Mathematics. 8. Cambridge University Press. ISBN 0-521-36764-6. https://books.google.com/books?id=yJwNrABugDEC&pg=PA123.