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By L.J. Jr. Ratliff

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Extra resources for Chain Conjectures in Ring Theory: An Exposition of Conjectures on Catenary Chains

Example text

4)), depth PI = n . 3) that it may be assumed that contains (R,M) depth q = n-! 1) = satisfies height p = i , then On the other hand, i (iii) = (i)] R [HMc, Proposition height PI = i depth p = depth pD - i = depth PI mcpil n so , and this holds. 3) depth Pk-i = a-I = depth Pk-I = d+k-I ~ d+h-i < a-i is a contradiction. 6). d. 7]° A diagram of the implications have been proved gram, in this chapter between is given on the next page. the numbers under a conjecture the numbers on the lines b e t w e e n implication is proved.

4) main REMARK. 3) Proof. d. 5). i ~ h . Repetitions . -. 1) clear height ... 4) so assume h > I . 1). assume statements depth p = h . 1). 10). 4) or the preceding the Catenary Chain Conjecture, paragraph) then it fol- that the H-Con- and the Normal Chain Conjecture also hold. 5) REMARK. , is either with then Proof. 2, pp. c. 9) and the in [N-6, Example domain with and that, exactly D/P has a domain of if R , then Hence, n £ [2,a+l} is catenary. 1) 1 , so ideal Hence, for each since there is a D/Q = (D/P)/(Q/P)) mcpil two = altitude for each height one prime and then R' R' Therefore, = (A.

3) If R is an H-local domain, then, for all maximal ideals M' in R' , R' M, is an H-d0main. is an H-ring. 39 Proof. 2) is level. , by holds and let R be an that RH If there are no height one maximal is level, by in assume b , S = R' so RH RH be a local domain such is an H-ring, H-local domain. 1). H-ring. 1), depth z = a} , b y IN-6, 2, p. 188]. 2). R' Let , let c E M' R' that = R) ideal in R' b and is an in B' H- , by and since H-ring, [N-6, Ex. 1) the same total quotient I B' I = ~[z E Spec R H ; and since if ; M' Ex.

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