# Chain Conjectures in Ring Theory: An Exposition of by L.J. Jr. Ratliff

By L.J. Jr. Ratliff

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C. H - s e m i - l o c a l d o m a i n that is integral over then, for each m a x i m a l ideal M in R , RM is an H-domain. 3) If R is a taut semi-local d o m a i n that is a finite in- tegral e x t e n s i o n o f a loca___~lsub-domain, Proof. so assume R R . 1) = let is integral over a local s u b d o m a i n in then A . such that , and let Then L l-c R be an H-domain. 7), H - s e m i - l o c a l d o m a i n that L o , and let M be a m a x i m a l is in all other m a x i m a l L = Lo + J , where J ideal ideals in is the J a c o b s o n is a local d o m a i n and it is readily seen that is a special e x t e n s i o n of L .

The W e a k Depth Conjecture local domain and that i_~s, i f holds; is such that p E Spec R that i_~s, i f height P = h such that R is a and depth height p = 1 and depth p ~ h . 6) (R,M) is a local domain, then the W e a k Depth Conjec- R[X](M,X ) The H-Qonjecture holds; that is, an H-local domain is catenary. 9) RSM. i domain and is an Chain Conjecture local domain domain R Rs H-domain, holds; satisfies The Normal Chain Conjecture gral closure o f ! c. c. Proof. c. For this, Then it may clearly be assumed P ~ Spec R , so R/P let R that then a Henselian be a Henselian a = altitude R > 1 .

20)], I showed that the Chain Conjecture holds for level Noetherian Hilbert domains, is satisfied: if the following condition if D is a level Noetherian Hilbert domain, is level, fo~ all b in the quotient field of D . then D[b] But this condition does not hold, as follows from K. 8). 7)), but it is known that it does not hold for at least some quasi-local domains. Namely, J. Sally showed in 1970, in [S], that there exists an integrally closed quasi-local domain that is not catenary, and I. 1) also showed this.