Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



$ H$-semilocal domains and altitude $ R[c/b]$

Author: L. J. Ratliff
Journal: Proc. Amer. Math. Soc. 64 (1977), 1-7
MSC: Primary 13C15; Secondary 13G05, 13B20
MathSciNet review: 0441951
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 R' of R if and only if every height one prime ideal in R' 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 [Enhancements On Off] (What's this?)

  • [1] J. T. Arnold, On the dimension theory of overrings of an integral domain, Trans. Amer. Math. Soc. 138 (1969), 313-326. MR 0238824 (39:188)
  • [2] P. Jaffard, Théorie de la dimension dans les anneaux de polynomes, Gauthier-Villars, Paris, 1960. MR 0117256 (22:8038)
  • [3] M. Nagata, Local rings, Interscience Tracts in Pure and Appl. Math., no. 13, Interscience, New York, 1962. MR 0155856 (27:5790)
  • [4] L. J. Ratliff, Jr., Characterizations of catenary rings, Amer. J. Math. 93 (1971), 1070-1108. MR 0297752 (45:6804)
  • [5] -, Chain conjecture and H-domains, Conf. on Commutative Algebra, Lecture Notes in Math., vol. 311, Springer-Verlag, Berlin and New York, 1973. MR 0337945 (49:2714)
  • [6] -, Conditions for $ \operatorname{Ker}(R[X] \to R[c/b])$ to have a linear base, Proc. Amer. Math. Soc. 39 (1973), 509-514. MR 0316442 (47:4989)
  • [7] -, Four notes on saturated chains of prime ideals, J. Algebra 39 (1976), 75-93. MR 0399072 (53:2923)
  • [8] -, The chain conjectures and valuation rings (90 page preprint).
  • [9] D. Rees, A note on valuations associated with a local domain, Proc. Cambridge Philos. Soc. 51 (1955), 252-253. MR 0070628 (17:10d)
  • [10] A. Seidenberg, On the dimension theory of rings. II, Pacific J. Math. 4 (1954), 603-614. MR 0065540 (16:441g)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13C15, 13G05, 13B20

Retrieve articles in all journals with MSC: 13C15, 13G05, 13B20

Additional Information

Keywords: Chain Conjecture, H-domain, integral closure, integral extension domain, Jacobson radical, polynomial ring, semilocal domain, simple extension ring, valuation ring
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society