bound on the Krull dimension of polynomial rings
If $A$ is a commutative ring, and $\mathrm{dim}$ denotes Krull dimension, then
$\mathrm{dim}(A)+1\le \mathrm{dim}(A[x])\le 2\mathrm{dim}(A)+1.$ |
It is known (see [Seid],[Seid2]) that for any $k\ge 0$ and $n$ with $k+1\le n\le 2k+1$, there exists a ring $A$ such that $dimA=k$ and $dimA[x]=n$.
References
- 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.
Title | bound on the Krull dimension of polynomial rings |
---|---|
Canonical name | BoundOnTheKrullDimensionOfPolynomialRings |
Date of creation | 2013-03-22 15:22:11 |
Last modified on | 2013-03-22 15:22:11 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 9 |
Author | mathcam (2727) |
Entry type | Theorem |
Classification | msc 13C15 |