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$ 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
DOI: https://doi.org/10.1090/S0002-9939-1977-0441951-9
MathSciNet review: 0441951
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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?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0441951-9
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

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