I am stuck at finding a ring $R$ with a chain of prime ideals of length $2008$. For reference,
Definition
A chain of prime ideals of length $n$ in a commutative ring $R$ is an increasing sequence $$P_0\subsetneq P_1 \subsetneq P_2 \subsetneq \cdots \subsetneq P_n \subsetneq R,$$
where $P_i$ is a prime ideal in $R$.
I realized that $R$ should not be PID since, in this case, maximum length of the chain is $1$. I would appreciate any help! Thanks in advance!
Best Answer
I guess $\Bbb R[x_1,\cdots,x_{7000}]$ works.