In commutative ring theory one considers the prime spectrum. This is a functorial
construction that associates a topological space with a ring. It serves at least two
important purposes: It is an invariant that encodes information about the ring. It helps
to translate algebraic information into geometric language, and vice versa. This
second aspect of prime spectra is the basis of their application in algebraic geometry
via schemes (cf. Grothendieck's EGA, or some introductory text about algebraic
geometry, such as …). Concerning the first aspect, the usefulness of spectra as
invariants depends to a large extent on understanding how properties of a ring
correspond to properties of its prime spectrum: Given a ring A with some arithmetic
property, does Spec A  have some corresponding topological property? Or, if
Spec A  has some particular topological property, does the ring A have some special
arithmetic properties? This kind of problem will be addressed in the present paper.
There are some well-known classical answers: If a ring is a domain then the prime
spectrum has a unique generic point. A ring has the property that, for every element a,
the element itself is a unit or 1  a is a unit if and only if there is just one closed point
in the prime spectrum (i.e., the ring is local).
Let P be a topological property that prime spectra may or may not have. We ask
whether the class of those rings whose prime spectrum has property P is
axiomatizable in the sense of first order model theory, using the language
L  ,,,0,1 of ring theory. There are many different possibilities for the property
P . Most of those that we consider are concerned with the space of maximal ideals or
with the space of minimal prime ideals, or with how these spaces sit inside the full
prime spectrum. Here is a small selection of properties that we study:
 The spectrum is normal, i.e., every prime ideal is contained in a unique maximal
ideal, or
 the spectrum is completely normal, i.e., the set of prime ideals that contain a given
prime ideal form a chain with respect to inclusion, or
 the spectrum consists of a single point, or
 the set of maximal points is a Hausdorff space, or
 the set of minimal points is a compact space.
We answer these and several other similar questions. For each question there are two
different variants: One may ask the question for all rings or only for reduced rings. If
the class of rings whose spectrum has property P is axiomatizable then the same is
clearly true for the class of reduced rings. This is the case, for example, if P says that
the spectrum is normal. On the other hand, if P means that the spectrum is
completely normal then neither the class of rings, nor the class of reduced rings is
axiomatizable. But if the prime spectrum has only one point then the answers are
different for all rings and for reduced rings: The class of reduced rings with only one
prime ideal is well-known to be the class of fields, which is clearly an axiomatizable
class. But the class of all rings with only one prime ideal is not axiomatizable, as we
shall prove.
At the end (see the table in section 12) we give an overview over the
axiomatizability of classes of rings defined in terms of properties of prime spectra.
Most of the answers are new; for completeness some well-known classical answers
are also included. Our answers are based upon a few key results and constructions.
The first one is Theorem ???, which shows that the rings with normal prime spectrum
form an axiomatizable class. Without much additional effort this leads to the fact that
the classes of rings whose space of maximal ideals is Hausdorff, or is Boolean, or is a
pro-constructible subspace of the full prime spectrum are all axiomatizable as well.
To show that the rings with completely normal prime spectrum is not axiomatizable
we refer to the fact (from model theory) that axiomatizable classes of structures are
closed under the formation of ultraproducts. We find a sequence An n• of rings
(with nilpotents) all of whose prime spectra have only one point, and we construct an
ultraproduct that has a factor ring k X,Y X,Y  , where k is a field. Thus the prime


spectrum of the ultraproduct contains Spec k X,Y X ,Y  , which is clearly not
completely normal. As complete normality is inherited by spectral subspaces one
concludes that the prime spectrum of the ultraproduct is not completely normal. The
construction can be modified to deal with reduced rings as well. The counterexamples
cover also rings with totally ordered prime spectra, with Boolean prime spectra, with
one-point prime spectra and with finite prime spectra.
These explanations account for most of the entries in the table. The most difficult
issue that remains is the question of compactness of the minimal prime spectrum. The
model theoretic method for proving non-axiomatizability is the same as before. But
the ring theoretic constructions we use are much more complicated. We modify and
extend a construction due to Quentel to produce a ring A with the following
properties:
 For every zero divisor a there is a finite set b1 ,K ,bn  Ann a  such that the
ideal a,b1 ,K ,bn   A is dense. The minimal size of such a set is denoted by
AS a  (the annihilator size of a).
 There are zero divisors with arbitrarily large annihilator size.
The ring itself has compact minimal prime spectrum. But there is an ultrapower that
has non-compact minimal prime spectrum.