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](https://en.wikipedia.org/wiki/Jacobson_ring), 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. 



Do commutative rings with "interesting" Jacobson radicals turn up "in nature"?
Rings similar to the ring $R$ as you consider above, "show up" in real algebraic geometry. If $k:=\mathbb{R}, A:=k[x]$ is the ring of polynomials with real coefficients, and $S\subseteq A$ is the set of polynomials with no real roots ($f(x):=x^2+1$ is such a polynomial), then $B:=S^{-1}A$ may be seen as the global sections of the "structure sheaf" of $X(k):=\mathbb{A}^1\_k(k):=Spec(A)(k)$ - the "affine real line" in the sense of real algebraic geometry (this is "vague").
You may construct a locally ringed space $(X(k), \mathcal{O}\_{X(k)})$, where $X(k)$ is the $k$-rational points of $X$ and where $\mathcal{O}:=\mathcal{O}\_{X(k)}$ i a sheaf on $X(k)$ with global sections equal to $B$.


**Example 1.** The rational function $f(x):=\frac{a(x)}{x^2+1}$ is a regular function on $X(k)$ and gives rise to a global section of $\mathcal{O}$ for any polynomial $a(x) \in A$. If $\mathfrak{m}$ is a maximal ideal in the localized ring $B$, it follows the residue field $\kappa(\mathfrak{m})\cong k$ is the field of real numbers.
Hence when you localize at $S$ you have removed all maximal ideals in $A$ with residue field the complex numbers.


**Example 2.** The global sections $\Gamma(X, \mathcal{O}\_X)=A$ is the polynomial ring, hence the real algebraic variety $X(k)$ has "more global sections" than the affine scheme $X$. Again this is "vague" since $X(k)$ is a subspace of $X$.


You may similarly construct the real projective line $\mathbb{P}(1):=\mathbb{P}^1\_k(k)$, its structure sheaf $\mathcal{O}\_{\mathbb{P}(1)}$
and the real affine space $\mathbb{A}(n)$ and
you may embed $\mathbb{P}(1)$ as a "closed subvariety" (in the sense of real algebraic geometry)


$$ \phi: \mathbb{P}(1) \rightarrow \mathbb{A}(3)$$


of affine $3$-space.


The above approach to "real algebraic geometry" uses the classical language of "schemes" and introduce a real algebraic variety using the $k$-rational points of the scheme $\mathbb{A}^1\_k$. There are other approaches and you find references at the zentralblatt site under "real algebraic geometry". The apporoach indicated above gives for any scheme $X$ of finite type over $k$, a "real algebraic variety" $(X(k), \mathcal{O}\_{X(k)})$, which is a locally ringed space where the structure sheaf $\mathcal{O}\_{X(k)}$ has "more" local sections. In the case of projective space $\mathbb{P}^n\_k(k)$ you get a space with properties similar to the underlying smooth manifold. You may (similar to the situation with differentiable manifolds and the Whitney embedding theorem) embed $\mathbb{P}^n\_k(k)$ into "real affine space" $\mathbb{A}^d\_k(k)$. In fact I believe there is an endofunctor $F$ of the category of schemes over $k$ with the property that for any projective scheme $X \subseteq \mathbb{P}^n\_k$ it follows the scheme $F(X):=X(k)$ is affine: There is a closed embedding $F(X) \subseteq \mathbb{A}^d\_k(k)$. The scheme $F(X)$ is no longer of finite type over $k$.


**Example 2.** A motivation for doing this is the study of the Weil restriction $W(X)$ of a complex projective manifold $X \subseteq \mathbb{P}^n$. In the case when the Weil restriction $W(X) \subseteq \mathbb{P}^m\_k$ is projective over $k$ you get in a functorial way an "affine real algebraic variety" $F(W(X))=Spec(B)$, and any complex holomorphic vector bundle $E$ on $X$ gives a finite rank projective $B$-module $E(k)$.


You may have heard of the "Jouanolou-Thomason trick", which to any quasi projective variety $X$ of finite type over a field $k$ associates an affine variety $Y$ of finite type over $k$ with $K\_i(Y)=K\_i(X)$. With the above construction you get a similar construction for complex projective varieties which is functorial.


