Summary
The purpose of this paper is to study the general properties of Hi-rings (a ring R is an Hi-ring in case, for every height i prime ideal p in R, height p+depth p=altitude R) and to link the results to Nagata's chain conjectures. For R a local domain with maximal ideal M equivalences are given to « R is an Hi-ring » in relationship to the rings R/p, where p is a prime ideal in R such that height p≤i; R[X 1 , …, Xn]MR[X 1 ,…,xn], where X 1 ,…,Xn are indeterminates; and R[c 1 /b,…, cj/b]MR[c 1/b,…,c j/b], where b, c 1 ,…,cj are analytically independent elements in R and j≤i. Also, there are equivalences of R[X](M,X), R[c/b](M,c/b), the completion of R, the Henselization of R and localities of R being Hi-rings in terms of conditions on R. These results then yield some new equivalences of the chain conjectures.
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Entrata in Redazione il 28 luglio 1977.
Most of the results in this paper are from the author's doctoral dissertation at the University of California, Riverside under the supervision of ProfessorL. J. Ratliff, Jr. and with the financial support of a National Science Foundation Traineeship.
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Pettit, M.E. Properties ofH i-Rings. Annali di Matematica 121, 1–23 (1979). https://doi.org/10.1007/BF02411990
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DOI: https://doi.org/10.1007/BF02411990