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bound on the Krull dimension of polynomial rings
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(Theorem)
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If  is a commutative ring, and
 denotes Krull dimension, then
It is known (see [Seid],[Seid2]) that for any  and  with
 , there exists a ring  such that  and
![$ \dim A[x]=n$](http://images.planetmath.org:8080/cache/objects/7196/l2h/img9.png "$ \dim A[x]=n$") .
- Seid
- A. Seidenberg, A note on the dimension theory of rings. Pacific J. of Mathematics, Volume 3 (1953), 505-512.
- Seid2
- A. Seidenberg, On the dimension theory of rings (II). Pacific J. of Mathematics, Volume 4 (1954), 603-614.
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Cross-references: ring, Krull dimension, commutative ring
There is 1 reference to this entry.
This is version 5 of bound on the Krull dimension of polynomial rings, born on 2005-06-29, modified 2006-12-12.
Object id is 7196, canonical name is BoundOnKrullDomainsOfPolynomialRings.
Accessed 1374 times total.
Classification:
| AMS MSC: | 13C15 (Commutative rings and algebras :: Theory of modules and ideals :: Dimension theory, depth, related rings ) |
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Pending Errata and Addenda
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![$\displaystyle \operatorname{dim}(A)+1\leq \operatorname{dim}(A[x])\leq 2\operatorname{dim}(A)+1.$](http://images.planetmath.org:8080/cache/objects/7196/l2h/img3.png "$\displaystyle \operatorname{dim}(A)+1\leq \operatorname{dim}(A[x])\leq 2\operatorname{dim}(A)+1.$"){kind=small}