$H$-semilocal domains and altitude $R[c/b]$
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- by L. J. Ratliff PDF
- Proc. Amer. Math. Soc. 64 (1977), 1-7 Request permission
Abstract:
It is shown that altitude $R[u] =$ altitude R holds for all u in the quotient field of a semilocal domain R such that $1/u$ is not in the Jacobson radical of the integral closure Rβ of R if and only if every height one prime ideal in Rβ has depth = altitude $R - 1$. Also, if (R,M) is a local domain, then every height one prime ideal p in $R[X]$ such that $p \subseteq (M,X)R[X]$ has depth = altitude R if and only if this holds for all such prime ideals which contain a linear polynomial.References
- Jimmy T. Arnold, On the dimension theory of overrings of an integral domain, Trans. Amer. Math. Soc. 138 (1969), 313β326. MR 238824, DOI 10.1090/S0002-9947-1969-0238824-3
- Paul Jaffard, ThΓ©orie de la dimension dans les anneaux de polynomes, MΓ©mor. Sci. Math., Fasc. 146, Gauthier-Villars, Paris, 1960 (French). MR 0117256
- Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
- L. J. Ratliff Jr., Characterizations of catenary rings, Amer. J. Math. 93 (1971), 1070β1108. MR 297752, DOI 10.2307/2373746
- Louis J. Ratliff Jr., Chain conjectures and $H$-domains, Conference on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972), Lecture Notes in Math., Vol. 311, Springer, Berlin, 1973, pp.Β 222β238. MR 0337945
- L. J. Ratliff Jr., Conditions for $\textrm {Ker}(R[X]\rightarrow R[c/b])$ to have a linear base, Proc. Amer. Math. Soc. 39 (1973), 509β514. MR 316442, DOI 10.1090/S0002-9939-1973-0316442-2
- L. J. Ratliff Jr., Four notes on saturated chains of prime ideals, J. Algebra 39 (1976), no.Β 1, 75β93. MR 399072, DOI 10.1016/0021-8693(76)90062-4 β, The chain conjectures and valuation rings (90 page preprint).
- D. Rees, A note on valuations associated with a local domain, Proc. Cambridge Philos. Soc. 51 (1955), 252β253. MR 70628, DOI 10.1017/s0305004100030164
- A. Seidenberg, On the dimension theory of rings. II, Pacific J. Math. 4 (1954), 603β614. MR 65540, DOI 10.2140/pjm.1954.4.603
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 64 (1977), 1-7
- MSC: Primary 13C15; Secondary 13G05, 13B20
- DOI: https://doi.org/10.1090/S0002-9939-1977-0441951-9
- MathSciNet review: 0441951