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.