dominates ; i.e., for each proper ideal I of A, is proper and for each maximal ideal of B, is maximal
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]
theorem, transition, algebra, theorem, transition, said, hold, between, commutative, rings, displaystyle, subset, displaystyle, dominates, displaystyle, each, proper, ideal, displaystyle, proper, each, maximal, ideal, displaystyle, mathfrak, displaystyle, math. In algebra the theorem of transition is said to hold between commutative rings A B displaystyle A subset B if 1 2 B displaystyle B dominates A displaystyle A i e for each proper ideal I of A I B displaystyle IB is proper and for each maximal ideal n displaystyle mathfrak n of B n A displaystyle mathfrak n cap A is maximal for each maximal ideal m displaystyle mathfrak m and m displaystyle mathfrak m primary ideal Q displaystyle Q of A displaystyle A length B B Q B displaystyle operatorname length B B QB is finite and moreover length B B Q B length B B m B length A A Q displaystyle operatorname length B B QB operatorname length B B mathfrak m B operatorname length A A Q Given commutative rings A B displaystyle A subset B such that B displaystyle B dominates A displaystyle A and for each maximal ideal m displaystyle mathfrak m of A displaystyle A such that length B B m B displaystyle operatorname length B B mathfrak m B is finite the natural inclusion A B displaystyle A to B is a faithfully flat ring homomorphism if and only if the theorem of transition holds between A B displaystyle A subset B 2 References edit Nagata 1975 Ch II 19 a b Matsumura 1986 Ch 8 Exercise 22 1 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 nbsp This algebra related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Theorem of transition amp oldid 1088553577, wikipedia, wiki, book, books, library,