What is a Hilbert Ring?

In a small way, the next post will use the concept of a Hilbert ring. It’s something I’ve talked about before on this blog somewhere, but here I will summarize the essential facts of Hilbert rings so that the next post can be completely self-contained.

Now, a Hilbert ring is one of those really neat ideas, because you can reduce the question “what the devil is the Jacobson radical?” to “are there any nilpotents?”. Moreover, Hilbert rings are very common. Let me explain.

We will stay entirely in the realm of commutative rings.

Definition. A $G$-ideal $I$ in a ring $R$ is a prime ideal such that the fraction field of $R/I$ is finitely generated as an $R/I$ algebra.

So, a $G$-ideal is a special kind of prime ideal. It turns out that an ideal $I$ is a $G$-ideal in a ring $R$ if and only if it can be written as $M\cap R$ where $M$ is a maximal ideal of $R[x]$. One fact that makes $G$-ideals so important is:

Theorem. In a commutative ring $R$, the nilradical of $R$ is the intersection of all the $G$-ideals of $R$.

Maximal ideals are of course examples of $G$-ideals, because the quotient of a ring by a maximal ideal is a field. This motivates the following definition:

Definition. A Hilbert ring is a commutative ring in which every $G$-ideal is maximal.

This definition wouldn’t be much use if Hilbert rings were rare. But, they are actually quite common. For example, the ring of integers $\Z$ is a Hilbert ring. Any field is of course a Hilbert ring. More importantly:

  1. The homomorphic image of a Hilbert ring is a Hilbert ring.
  2. A ring $R$ is a Hilbert ring if and only if $R[x]$ is a Hilbert ring.

Hence, in particular, the finitely generated $\Z$ algebras and $k$-algebras where $k$ is a field are all Hilbert rings. This includes the coordinate rings of affine varieties over a field. So, for any of these Hilbert rings, the Jacobson radical is the nilradical!

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