At lots of places it is stated, that Hilbert's Nullstellensatz is well understood as theorem about more general Jacobson rings. Namely
(When $R$ is a Jacobson ring and $S$ finitely generated $R$-algebra, then $S$ is also Jacobson.) If $M$ is a maximal ideal of $S$, then the pullback ideal $N=M\cap R$ of $R$ is also maximal and the field $S/M$ is finite extension over the field $R/N$.
Why is this considered to be generalize Nullstellensatz? Isn't that mainly a statement about restrictions for existence of solutions, or more precise characterization of the Galois connection in question.
The general version statement is in my opinion analogous to Zariski's lemma, which is used in the first step of the usual Nullstellensatz' proof.
It's very well possible it's called Nullstellensatz even though the analogy is not a strong one, because the Zariski's lemma is the crucial ingredient in the proof. However, I'm not writing here to rant about the name. I want to make sure I'm not missing something: Are the other parts of Nullstellensatz deducible from the general form or is there more to the theory of Jacobson rings, that would generalize the usual case?
Best Answer
Note that the classical Nullstellensatz in $k[x_1, \dotsc, x_n]$ states $$\sqrt{\mathfrak a} = \mathcal I(\mathcal V(\mathfrak a)) = \bigcap\limits_{\mathfrak a \subset \mathfrak m} \mathfrak m,$$ with the intersection being taken over all maximal ideals containing $\mathfrak a$.
In any ring, we have a formal Nullstellensatz: $$\sqrt{\mathfrak a} = \bigcap\limits_{\mathfrak a \subset \mathfrak p} \mathfrak p,$$ with the intersection being taken over all prime ideals containing $\mathfrak a$. Note that the most proofs of this fact are actually modern versions of the Rabinowitsch trick (Rabinowitsch trick is just localizing, not so tricky from the modern point of view ;) ).
In a Jacobson ring, we have $\bigcap\limits_{\mathfrak a \subset \mathfrak p} \mathfrak p = \bigcap\limits_{\mathfrak a \subset \mathfrak m} \mathfrak m$, i.e. the formal Nullstellensatz boils down to the formulation of the classical Nullstellensatz. This is why a result about many rings being Jacobson is called a generalization of the Nullstellensatz.