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Maximal Ideal Transforms of Noetherian Rings

1976; American Mathematical Society; Volume: 54; Issue: 1 Linguagem: Inglês

10.2307/2040746

1088-6826

J. Matijevic,

Rings, Modules, and Algebras

Let J? be a commutative Noetherian ring with unit.Let T be the set of all elements of the total quotient ring of R whose conductor to R contains a power of a finite product of maximal ideals of R. If A is any ring such that R C A C T, then A/xA is a finite R module for any non-zerodivisor x in R. It follows that if, in addition, R has no nonzero nilpotent elements, then any ring A such that R C A C T is Noetherian.Let fi be a commutative Noetherian domain with unit of Krull dimension one.The Krull-Akizuki theorem [6, Theorem 33.2] states that if Pis an integral domain containing R and contained in a finite algebraic extension of the quotient field of R then P is Noetherian.By adjoining a finite number of elements to R and letting this new ring be called R, one proves the theorem by proving the following: Any ring A between a Noetherian domain R of Krull dimension one and its quotient field is Noetherian.This is equivalent to showing that if x is any nonzero element of R then A/xA is a finite R module.We shall restate this reduction of the Krull-Akizuki theorem in such a way that the final statement is true for any Noetherian ring of any Krull dimension.In order to do this we first characterize the relationship between a one dimensional Noetherian domain R and its quotient field P. If y belongs to P, then its conductor to R, or its denominator ideal contains some powered product of a finite number of maximal ideals of R. In the special case that R has precisely one maximal ideal M, T is the set of all elements whose conductor to R contains a power of M.If B is any commutative Noetherian ring with unit and / is any ideal of B that contains a non-zero-divisor, the Ptransform of B is defined to be the set of all elements of the total quotient ring whose conductor to B contains a power of I.The Ptransform is a ring between B and its total quotient ring.If B is a Noetherian domain of Krull dimension one and B has one maximal ideal M, then the quotient field is the A/-transform of B. Now if R is any commutative Noetherian ring with unit, we call P the global transform of R if P is the set of all elements of the total quotient ring whose conductor to R contains a power of a finite product of maximal ideals of R. If M is any maximal ideal of R, TM = P ®R RM, where RM is the localization of R at M, is the A/fi ^-transform of RM.Also P contains the M-transform of R for M any maximal ideal of R. If R is a domain of Krull dimension one, we