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Let $R$ be a commutative ring. Let's say that the Jacobson radical $J(R)$ of $R$ is uninteresting if

  1. $J(R)$ coincides with the nilradical, or

  2. $J(R)$ is the intersection of a finite number of maximal ideals.

It seems as if most rings used in algebraic geometry have uninteresting Jacbson radical:

  • Every finitely-generated commutative algebra over a field or over a Dedekind domain is Jacobson, so its Jacobson radical coincides with its nilradical, and so is uninteresting by (1).

  • Every local or semilocal commutative ring has finitely many maximal ideals, and so has uninteresting Jacobson radical by (2).

For a nonuninteresting example, take the localization $R = \mathbb Z[x]_S$ where $S = \{f(x) \in \mathbb Z[x] \mid f(0) = 1\}$. The maximal ideals of $R$ are of the form $(p,x)$ where $p \in \operatorname{Spec} \mathbb Z$. So the Jacobson radical is $(x)$, which is not uninteresting.

But this example seems rather artificial to me; for example I don't know anywhere a ring like this would show up in algebraic geometry.

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    $\begingroup$ Sure, tons of these arise when completing along a non-maximal non-nilpotent ideal. Let $O$ be a complete local noetherian ring with non-nilpotent maximal ideal $m$ and $m$-adically complete $O[x_1,\dots,x_n]$ with $n>0$. We get the ring $O\{x_1,\dots,x_n\}$ of formal power series $\sum a_I x^I$ whose coefficients $a_I$ tend to 0 in $O$ as $|\!|I|\!| \to \infty$. The Jacobson radical is $m\{x_1,\dots,x_n\}$, which is not the nilradical (since $m$ isn't nilpotent), whereas finite intersections of maximal ideals correspond to finite sets of closed points in $\mathbf{A}^n_{O/m}$ (with $n>0$). $\endgroup$ – nfdc23 Mar 7 '18 at 6:30
  • $\begingroup$ Oh wow -- I had discounted such examples out of the misconception that the completion of a ring at a (prime) ideal always factors through the localization at that ideal (and hence is local, with uninteresting Jacobson radical), which I now see is very much false when the ideal is not maximal. Rings like this are primarily important in deformation theory, right? $\endgroup$ – Tim Campion Mar 7 '18 at 6:59

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