Jacobson ring

A commutative ring with unit element in which any prime ideal is the intersection of the maximal ideals containing it, i.e. a ring any integral quotient ring of which has a zero Jacobson radical. For instance, any Artinian ring, any ring of integers (in general, any Dedekind ring which is not semi-local) or any absolutely-flat ring is a Jacobson ring. On the contrary, a local non-Artinian ring is not a Jacobson ring.

If $A$ is a Jacobson ring and $B$ is an integral $A$-algebra or an $A$-algebra of finite type, then $B$ will be a Jacobson ring; in particular, a quotient ring of a Jacobson ring is a Jacobson ring. A ring of polynomials in a finite number of variables over a field $K$ is a Jacobson ring; if the number of variables is infinite, the answer will depend on relations between the number of variables and the cardinality of the field $K$. A ring $A$ is a Jacobson ring if the space of maximal ideals of $A$ is quasi-homeomorphic to the spectrum $\mathrm{Spec}(A)$; this definition leads to the concept of a Jacobson scheme.

References

[1]	N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)

Comments

Quite generally, a (non-commutative) ring $A$ is a Jacobson ring if every prime ideal is an intersection of primitive ideals or, equivalently, if every prime factor ring $A/\mathfrak{p}$, $\mathfrak{p}$ a prime ideal, has zero Jacobson radical. Here an ideal $\mathfrak{q} \subset A$ is primitive if the quotient $A/\mathfrak{q}$ is a primitive ring.

An algebra $A$ over a field $k$ which is a Jacobson ring and for which moreover every primitive ideal has finite codimension is sometimes called a Hilbert algebra. The theorem that a finitely-generated polynomial-identity algebra (cf. PI-algebra) over a field is a Hilbert algebra, is a non-commutative generalization of the Hilbert Nullstellen Satz, [a1], Chapt. V. This notion of a Hilbert algebra should not be confused with the one defined in the article Hilbert algebra, which refers to an algebra provided with an involution and an inner product satisfying certain properties.

References

[a1]	C. Procesu, "Rings with polynomial identities" , M. Dekker (1973)
[a2]	J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987)