A prime ideal theorem is a theorem stating that every proper ideal is contained in some prime ideal. A prime ideal theorem is typically equivalent to the ultrafilter principle (UF), a weak form of the axiom of choice (AC).
We say ‘a’ prime ideal theorem (PIT) instead of ‘the’ prime ideal theorem, since we have not said what the ideals are in. We list some representative examples of prime ideal theorems, all of which are equivalent to (UF) in ZF or even in BZ (bounded Zermelo set theory):
The PIT for (commutative) rings.
The PIT for (bounded) distributive lattices.
The PIT for Boolean algebras.
The PIT for rigs, which subsumes all of the above. In fact one can generalize quite a bit further, as we indicate below when we discuss prime element theorems.
The usual way to prove a prime ideal theorem is with the help of Zorn's Lemma, unless one is specifically trying to use something weaker like UF (as we do in this article). In fact Zorn's lemma can be used to prove a maximal ideal theorem, which is stronger: in all the usual examples maximal ideals are prime so that a prime ideal theorem becomes a corollary.
There are stronger forms of prime ideal theorems which assert the existence of a prime ideal that does not intersect a given multiplicatively closed subset . However, these can often be reduced to an ordinary PIT, by first localizing or inverting the elements in , and then applying a PIT to the localization.
Prime ideal theorems may be arranged as a sequence of results and techniques that gradually increase in sophistication; we sketch one possible development here, filling in details in the remainder of the article.
A first easy wave of results concerns the equivalence between the Boolean PIT (or BPIT) and UF. While these are easy, the BPIT is a theoretical underpinning of fundamental techniques such as the compactness theorem for first-order logic, which leads to the next rung on the ladder.
The next step derives the PIT for distributive lattices as a simple application of the compactness theorem. In effect we write down a simple propositional theory where a model is tantamount to a prime ideal in a given distributive lattice. Part of the work involves checking finite satisfiability of the theory, where we may invoke a simple Stone duality between finite distributive lattices and finite posets to get the job done.
With the PIT for distributive lattices in hand, we prove a key result due to Banaschewski, which may be summarized roughly as saying that a (nontrivial) compact frame admits a prime element. This is an early result in Stone Spaces, related to the spatiality of compact regular locales.
Banaschewski’s lemma is a key of entry into prime ideal theorems of general type. The general idea is that ideals in a monoid (now in suitably nice monoidal categories) tend to form compact quantales, whose prime elements correspond to prime ideals. One then invokes a simple construction which associates to each compact quantale a quotient compact frame, in such a way that the existence of a prime element in the quantale is reduced to existence of a prime element in the compact frame à la Banaschewski’s lemma. As promised, this gives prime ideal theorems of fairly general type (call them collectively “GPIT”).
As suggested earlier, any one of these prime ideal theorems implies the BPIT as a special case, and so we come full circle:
We now turn to details.
The Boolean prime ideal theorem or BPIT is equivalent to the ultrafilter principle UF.
The reasoning may be summarized as follows: the ultrafilter principle implies the Tychonoff theorem for compact Hausdorff spaces; see this Remark and the argument immediately preceding it. The Tychonoff theorem for compact Hausdorff spaces in turn implies that every Boolean ring has a maximal (and therefore prime) ideal; see here. Consequently, for any ideal of a Boolean algebra , the quotient Boolean ring has a prime ideal , and the pullback of the quotient map produces a prime ideal in which contains a given ideal , thus proving the BPIT from UF.
(In passing, one may note that prime ideals in Boolean algebras (or Boolean rings) are the same as maximal ideals. For, the corresponding Boolean quotient ring is an integral domain only if it is the field , since idempotency of elements yields and then or assuming no zero divisors.)
Finally, maximal ideals are complementary to ultrafilters (see here). So UF, which says that every filter in is contained in an ultrafilter, translates into a special case of the BPIT: every ideal of (consisting of negations of elements of a corresponding filter) is contained in a prime/maximal ideal. This brings us full circle: BPIT implies UF implies Tychonoff(CH) implies BPIT.
Now let us prove that the Tychonoff theorem for CH spaces implies the PIT for distributive lattices : that any proper ideal of is contained in a prime ideal of . By passing to the quotient lattice , it suffices to show that has a prime ideal , since the inverse image of a prime ideal along the quotient map is again prime.
(UF) Any nontrivial distributive lattice has a prime ideal .
To prove this, we write down a propositional theory of “a prime ideal of ” (and consider its Lindenbaum algebra). Form a free Boolean algebra freely generated by the underlying set of . So each corresponds to a generator , which we will think of as standing for a proposition “”. The conditions that enforce “ is a prime ideal of ” are the axioms of our theory:
( is closed under finite joins) , .
( is downward closed) For each , .
( is proper) .
(primality condition) .
These axioms (certain elements of ) then generate a filter . As soon as we know is proper (“finite satisfiability of the theory”), the BPIT ensures that is contained in an ultrafilter , corresponding to a model or a Boolean ring homomorphism out of the Lindenbaum algebra. For that model, the collection then forms a prime ideal of , as desired. So:
(The filter is proper.) To see that the filter generated by finitely many axioms does not contain , consider the sublattice of generated by the finitely many elements named in atomic subformulas of such axioms. A finitely generated distributive lattice is finite (indeed, by Stone duality, the free distributive lattice on a finite number or elements is isomorphic to the poset of downward-closed subsets of the poset ). So all we need to do is show that any nontrivial finite distributive lattice has a prime ideal. This also follows from Stone duality: a distributive lattice homomorphism corresponds to a point of the Stone dual , and if is nontrivial then must be inhabited by a point , so lattice homomorphisms exist and the kernel of such is a maximal and therefore prime ideal.
During the 1980’s, Banaschewski discovered the following result (a breakthrough in the study of prime ideal theorems, according to Marcel Erné). We state and prove the lemma for now, and show how it is used in later sections.
Here “nontrivial” means : distinct top and bottom elements. This is needed because prime ideals are required to be proper (see too simple to be simple). An element is prime if the principal ideal it generates is prime, i.e., if implies or . In a distributive lattice, this is the same as saying implies or (meet-irreducibility).
First we observe that a coproduct of nontrivial bounded distributive lattices (in the category of bounded distributive lattices) is also nontrivial. Again this is a kind of finite satisfiability result: if such a coproduct were trivial, i.e., if were satisfied in the coproduct (constructed syntactically as the free distributive lattice modulo equations that identify finite meets and joins in each with formal meets and joins in ), then would be derivable from finitely many equations involving finitely many elements in the , and hence would be derivable in a finite coproduct of finite bounded nontrivial distributive lattices generated by these elements. This cannot happen, by considering their Stone duality with finite posets and products thereof: finite products of inhabited finite posets are also inhabited.
With that in hand, consider the coproduct of all the principal filters of , taken in the category of bounded distributive lattices; let denote the coproduct inclusion. has a prime ideal by the PIT for distributive lattices; the pullback is a prime ideal of . Of course by properness of prime ideals.
Put . This is a dcpo by completeness of and compactness of ; since proper ideals are directed subsets, we see that admits a sup for each . We have for all . By the Bourbaki-Witt fixed point theorem, the inflationary operator has a fixed point, say . For that , note that is just .
So is a prime ideal in . By meet-irreducibility, this means is a prime element in .
When the statement is about compact frames, it can be rephrased as saying that every nontrivial compact locale has at least one point; compare Stone Spaces, Lemma 1.9.
To show how Banaschewski’s lemma is applied, we consider for example the prime ideal theorem for (possibly noncommutative) rings. We follow the common convention that rings have units.
Let be a ring, and let be the sup-lattice of two-sided ideals of . Under multiplication of ideals , we obtain a quantale structure on whose multiplicative unit is the top element . In general we will call a quantale affine (or semicartesian) if its unit is the top element.
The top element of is also a compact element. In general, and generalizing a definition from the theory of frames/locales, we will say that a quantale is compact if its top element is a compact element.
Finally, in a quantale , we say an element is prime if for all we have implies or . Prime elements in are precisely prime ideals of .
(ZF + UF) Every nontrivial compact affine quantale has a prime element.
The strategy (following Paseka) is to reduce the claim to an application of Banaschewski’s lemma.
We begin by introducing an equivalence relation on : say if .
The relation is a quantale congruence.
First we show that respects the quantale multiplication . Suppose and . So for any , if , it follows that and , whence and . Then
and similarly implies . So .
Now we show that respects joins, i.e., if for all , then . Suppose . Since is compact, there is a finite set of the , say , such that . From and , we derive
and we proceed by induction to show , which implies . Similarly, implies , and so we are done.
The quantale formed as the quotient is a nontrivial compact frame.
Clearly implies . But also if , then
and thus for all . Being an idempotent affine quantale, is a frame.
We have since ( iff ) fails for , under the assumption is nontrivial. So is nontrivial. The same proof shows that for any , we have if .
To show is compact, suppose is a family in such that , in other words such that for all . Setting and applying compactness of , there is a finite subfamily such that , and then which proves compactness of .
The quantale quotient map preserves arbitrary sups and thus has a right adjoint , by the poset form of the adjoint functor theorem. If we denote the -class of an element by , then for the element is given by the explicit formula
Given a regular epi in the category of quantales, with right adjoint , if is prime in , then is prime in .
For elements we have
which completes the proof.
The proof of Theorem 3 is immediate:
(PIT for rings) Every unital ring has a prime ideal under ZF + (UF).
See also Banaschewski-Harting. We remark that the nucleus on formed by the composition has the following properties:
These properties are all that is needed to verify that the fixed points of (corresponding to the quotient ) forms a compact frame to which we can apply Banaschewski’s lemma. For example, the nucleus obtained by taking the Jacobson radical of an ideal has these properties, as do other radicals considered in noncommutative ring theory (Levitski radical, Brown-McCoy radical).
In fact the same methods show that we can strengthen Theorem 3 to get what we really mean by a PIT:
Let be a nontrivial compact affine quantale, and let be any proper element, i.e., assume . Then there exists a prime element such that .
Let be the nucleus corresponding to the congruence (i.e., the regular epi ). Since , we also have (i.e., ), as shown in the proof of Proposition 1. So and the upwards-closed set are both nontrivial compact affine frames, and there is a prime element of . We have a regular epi of quantales defined as the composite
(here using the fact that preserves finite meets, since is a distributive lattice). The right adjoint of is the inclusion , and it follows from Lemma 2 that is a prime of such that . Since , we conclude .
Now we consider more general settings in which we can speak of (two-sided) ideals and prime ideals, and to which we can apply Theorem 3.
For example, let be a complete, cocomplete biclosed monoidal category (e.g., a cosmos). If is a monoid object in , then we can speak of ideals in , and typically will inherit good properties from . If for example is regular and well-powered, then forms a quantale under multiplication of ideals, as discussed here.
If we further assume that the monoidal unit of is compact (i.e., finitely presented), then the top element of generated by the monoid unit will be a compact element. Then, defining a prime ideal in as usual to be a prime element in , we deduce existence of prime ideals from Theorem 3.
Here is a sample result along these lines:
Let be a commutative Lawvere theory. Then the category of algebras is a well-powered regular cosmos in which the monoidal unit is compact. Consequently, for every monoid of and every proper ideal , there is a prime ideal containing (by applying Corollary 2 to ).
(PIT for rigs) Every proper ideal in a rig is contained in a prime ideal.