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Jaffard ring
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(Definition)
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Let $\operatorname{dim}$ denote Krull dimension. A Jaffard ring is a ring $A$ for which $\operatorname{dim}(A[x])=\operatorname{dim}(A)+1$ (compare to the bound on the Krull dimension of polynomial rings). Such a ring is said to be Jaffardian.
Since this condition holds for Noetherian rings, every Noetherian ring is Jaffardian. Examples of rings that are not Jaffardian are thus relatively difficult to come by, since we are already forced to search exclusively in the realm of non-Noetherian rings. The first example of a non-Jaffardian ring seems to have been found by A. Seidenberg [Seid]: the subring of $\ol{\mathbb{Q}}[[T]]$ consisting of power series whose constant term is rational.
A Jaffard domain is a Jaffard ring which is also an integral domain.
- Seid
- A. Seidenberg, A note on the dimension theory of rings. Pacific J. of Mathematics, Volume 3 (1953), 505-512.
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"Jaffard ring" is owned by mathcam.
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| Also defines: |
Jaffard domain, Jaffard |
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Cross-references: integral domain, rational, constant term, power series, subring, examples of rings, noetherian rings, bound on the Krull dimension of polynomial rings, ring, Krull dimension
This is version 1 of Jaffard ring, born on 2005-06-29.
Object id is 7197, canonical name is JaffardRing.
Accessed 3260 times total.
Classification:
| AMS MSC: | 13F05 (Commutative rings and algebras :: Arithmetic rings and other special rings :: Dedekind, Prüfer and Krull rings and their generalizations) |
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Pending Errata and Addenda
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