In algebra, the theorem of transition is said to hold between commutative rings \( A\subset B \) if[1][2]
(i) B dominates A {\displaystyle A} A; i.e., for each proper ideal I of A, \( {\displaystyle IB} \) is proper and for each maximal ideal \( {\displaystyle {\mathfrak {n}}} \) of B, \( {\displaystyle {\mathfrak {n}}\cap A} \) is maximal
(ii) for each maximal ideal \( \mathfrak m \) and \( \mathfrak m \)-primary ideal Q of A, length \( {\displaystyle \operatorname {length} _{B}(B/QB)} \) is finite and moreover
\( {\displaystyle \operatorname {length} _{B}(B/QB)=\operatorname {length} _{B}(B/{\mathfrak {m}}B)\operatorname {length} _{A}(A/Q).} \)
Given commutative rings \( A\subset B \) such that B dominates A and for each maximal ideal \( \mathfrak m \) of } A such that length \( {\displaystyle \operatorname {length} _{B}(B/{\mathfrak {m}}B)} \) is finite, the natural inclusion \( A\to B \) is a faithfully flat ring homomorphism if and only if the theorem of transition holds between \( A\subset B \).[2]
References
Nagata, Ch. II, § 19.
Matsumura, Ch. 8, Exercise 22.1.
Nagata, Local Rings
Matsumura, Hideyuki (1986). Commutative ring theory. Cambridge Studies in Advanced Mathematics. 8. Cambridge University Press. ISBN 0-521-36764-6. MR 0879273. Zbl 0603.13001.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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