# Density of maximal spectrum

It's well known that for algebraically closed field $k$ maximal spectrum of finitely generated $k$-algebra is everywhere dense in whole spectrum of this algebra.

What can be said in the case of non-algebraically closed field, or even if $k$ is not a field, $\mathbb{Z}$, for instance?

Claim: For a ring $R$, its maximal spectrum is dense in its whole spectrum if and only if its Jacobson radical $J(R)$ is equal to its nilradical $N(R)$.

Proof: The max spectrum of $R$ is dense in the spectrum if and only if, for all $f \in R$, either $D(f) = \varnothing$ or there exists a maximal ideal $M$ such that $M \in D(f)$. That is, either $f$ is contained in all prime ideals of $R$ (so that $D(f) = \varnothing$) or there exists a maximal ideal $M$ with $f \notin M$, or equivalently, either $f \in N(R)$ or $f \notin J(R)$. Because $N(R) \subseteq J(R)$, this dichotomy is finally equivalent to $N(R) = J(R)$ as claimed. $\square$

The well known case that you cite for a finitely generated algebra over an algebraically closed field follows from the fact that $k[x_1, \dots, x_n]$ is a Jacobson ring, so that every homomorphic image thereof is also Jacobson. In fact, since polynomial rings over any Jacboson ring are Jacobson, and because fields and the ring of integers are Jacobson rings, we can see immediately that every finitely generated $k$-algebra has the property that you desire when $k$ is any field (not necessarily algebraically closed) or when $k = \mathbb{Z}$.

• Thank you for the answer, but I have one detail unclear: how can be shown that polynomial ring over Jacobson ring is Jacobson?
– user165101
Commented Nov 13, 2014 at 19:19
• You should be able to find a proof of this in any textbook that deals with Jacobson rings (also known as "Hilbert rings"). Do you have some favorite books on commutative algebra or ring theory that you study from? I can try to point you to a reference. Commented Nov 13, 2014 at 20:03
• I'm using book of Atiyah and Macdonald but I'm not sure it's stated there.
– user165101
Commented Nov 13, 2014 at 20:26
• I see. You can find proofs in section 4.5 (on the Nullstellensatz) of Eisenbud's book, or section 1-3 of Kaplansky's Commutative Rings book. Commented Nov 13, 2014 at 20:51
• Or if it's not so easy for you to locate either book, see Theorem 15.26 of the following notes: web.mit.edu/18.705/www/13Ed-2up.pdf Commented Nov 13, 2014 at 20:52