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Theorem of transition

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In algebra, the theorem of transition is said to hold between commutative rings if[1][2]

  1. dominates ; i.e., for each proper ideal I of A, is proper and for each maximal ideal of B, is maximal
  2. for each maximal ideal and -primary ideal of , is finite and moreover

Given commutative rings such that dominates and for each maximal ideal of such that is finite, the natural inclusion is a faithfully flat ring homomorphism if and only if the theorem of transition holds between .[2]

Notes

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  1. ^ Nagata 1975, Ch. II, § 19.
  2. ^ a b Matsumura 1986, Ch. 8, Exercise 22.1.

References

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  • Nagata, M. (1975). Local Rings. Interscience tracts in pure and applied mathematics. Krieger. ISBN 978-0-88275-228-0.
  • 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.