An example of commutative ring which has conditions regarding Jacobson radical

Question

First of all, I use the following notations for ring $R$.
$J(R)$ is Jacobson radical of $R$.
$N(R)$ is nilradical of $R$.

Next, I say my question.

I want to find a commutative ring $R$ which has following two properties.
$(1)$ $J(R) \neq N(R)$.
$(2)$ Krull dimension of $R/J(R)$ is not $0$.

Can you construct a commutative ring $R$ satisfying $(1)$ and $(2)$ ?

Answer 1

Let $R = D[[x]]$ where $D$ is a Dedekind domain, for example $\mathbb{Z}$ or $F[x]$ for $F$ a field. $R$ is an integral domain, so its nilradical is trivial. Its Jacobson radical is $J(R) = (x)$, and $R/J(R) \cong D$ has Krull dimension $1$.