# bound on the Krull dimension of polynomial rings

If $A$ is a commutative ring, and $\operatorname{dim}$ denotes Krull dimension, then

 $\displaystyle\operatorname{dim}(A)+1\leq\operatorname{dim}(A[x])\leq 2% \operatorname{dim}(A)+1.$

It is known (see [Seid],[Seid2]) that for any $k\geq 0$ and $n$ with $k+1\leq n\leq 2k+1$, there exists a ring $A$ such that $\dim A=k$ and $\dim A[x]=n$.

## 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 BoundOnTheKrullDimensionOfPolynomialRings 2013-03-22 15:22:11 2013-03-22 15:22:11 mathcam (2727) mathcam (2727) 9 mathcam (2727) Theorem msc 13C15