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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 $(2)$ ?

user26857
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神宮寺春姫
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  • Try, $K[x, y]{z, w}/w^2$, The power series in variables $z, w$ ofer the polynomial ring in the variables $x, y$ modulo the ideal generated by the square of $w$ – R.C.Cowsik Feb 27 '19 at 05:18

1 Answers1

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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 $1$.

Qiaochu Yuan
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