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prime ideal theorem

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Idea

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 𝔪\mathfrak{m}. However, these can often be reduced to an ordinary PIT, by first localizing or inverting the elements in 𝔪\mathfrak{m}, and then applying a PIT to the localization.

A ladder of prime ideal theorems

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:

UFBPITPIT(DistLat)BanaschewskiGPITBPITUF.UF \Rightarrow BPIT \Rightarrow PIT(DistLat) \Rightarrow Banaschewski \Rightarrow GPIT \Rightarrow BPIT \Rightarrow UF.

We now turn to details.

Boolean PIT and the ultrafilter principle

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 BB has a maximal (and therefore prime) ideal; see here. Consequently, for any ideal II of a Boolean algebra BB, the quotient Boolean ring B/IB/I has a prime ideal PP, and the pullback q 1(P)Bq^{-1}(P) \subseteq B of the quotient map q:BB/Iq: B \to B/I produces a prime ideal in BB which contains a given ideal II, thus proving the BPIT from UF.

(In passing, one may note that prime ideals in Boolean algebras (or Boolean rings) BB are the same as maximal ideals. For, the corresponding Boolean quotient ring is an integral domain only if it is the field /(2)\mathbb{Z}/(2), since idempotency of elements yields e(1e)=0e(1-e) = 0 and then e=0e = 0 or e=1e = 1 assuming no zero divisors.)

Finally, maximal ideals are complementary to ultrafilters (see here). So UF, which says that every filter in B=PXB = P X is contained in an ultrafilter, translates into a special case of the BPIT: every ideal of PXP X (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.

BPIT implies prime ideal theorem for distributive lattices

Now let us prove that the Tychonoff theorem for CH spaces implies the PIT for distributive lattices DD: that any proper ideal II of DD is contained in a prime ideal of DD. By passing to the quotient lattice D/ID/I, it suffices to show that D/ID/I has a prime ideal PP, since the inverse image ϕ 1(P)\phi^{-1}(P) of a prime ideal PP along the quotient map ϕ:DD/I\phi: D \to D/I is again prime.

Theorem

(UF) Any nontrivial distributive lattice DD has a prime ideal PP.

To prove this, we write down a propositional theory of “a prime ideal PP of DD” (and consider its Lindenbaum algebra). Form a free Boolean algebra Bool(UD)Bool(U D) freely generated by the underlying set UDU D of DD. So each xUDx \in U D corresponds to a generator P xBool(UD)P_x \in Bool(U D), which we will think of as standing for a proposition P x=P_x = xPx \in P”. The conditions that enforce “PP is a prime ideal of DD” are the axioms of our theory:

  • (PP is closed under finite joins) P 0P_0, P aP bP abP_a \wedge P_b \Rightarrow P_{a \vee b}.

  • (PP is downward closed) For each bUDb \in U D, P aP abP_a \Rightarrow P_{a \wedge b}.

  • (PP is proper) ¬P 1\neg P_1.

  • (primality condition) P ab(P aP b)P_{a \wedge b} \Rightarrow (P_a \vee P_b).

These axioms (certain elements of Bool(UD)Bool(U D)) then generate a filter \mathcal{F}. As soon as we know \mathcal{F} is proper (“finite satisfiability of the theory”), the BPIT ensures that FF is contained in an ultrafilter 𝒰\mathcal{U}, corresponding to a model or a Boolean ring homomorphism Bool(UD)/2Bool(U D)/\mathcal{F} \to \mathbf{2} out of the Lindenbaum algebra. For that model, the collection {aD:P a𝒰}\{a \in D: P_a \in \mathcal{U}\} then forms a prime ideal PP of DD, as desired. So:

Proof

(The filter \mathcal{F} is proper.) To see that the filter generated by finitely many axioms does not contain 0Bool(UD)0 \in Bool(U D), consider the sublattice of DD generated by the finitely many elements aa 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 nn or elements is isomorphic to the poset of downward-closed subsets of the poset 2 n\mathbf{2}^n). So all we need to do is show that any nontrivial finite distributive lattice LL has a prime ideal. This also follows from Stone duality: a distributive lattice homomorphism L2L \to \mathbf{2} corresponds to a point of the Stone dual SS, and if LL is nontrivial then SS must be inhabited by a point xx, so lattice homomorphisms L2L \to \mathbf{2} exist and the kernel of such is a maximal and therefore prime ideal.

Banaschewski’s lemma

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.

Theorem

(Banaschewski’s lemma) Every nontrivial distributive complete lattice LL whose top element 11 is compact, e.g., a compact frame, has a prime element.

Remarks

Here “nontrivial” means 010 \neq 1: distinct top and bottom elements. This is needed because prime ideals are required to be proper (see too simple to be simple). An element pp is prime if the principal ideal it generates is prime, i.e., if abpa \wedge b \leq p implies apa \leq p or bpb \leq p. In a distributive lattice, this is the same as saying p=abp = a \wedge b implies p=ap = a or p=bp = b (meet-irreducibility).

Proof

First we observe that a coproduct of nontrivial bounded distributive lattices {X i} iI\{X_i\}_{i \in I} (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 0=10 = 1 were satisfied in the coproduct (constructed syntactically as the free distributive lattice F=F( iX i)F = F(\sum_i X_i) modulo equations that identify finite meets and joins in each X iX_i with formal meets and joins in FF), then 0=10 = 1 would be derivable from finitely many equations involving finitely many elements x ix_i in the X iX_i, and hence 0=10 = 1 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 M= aLaM = \sum_{a \in L} \uparrow a of all the principal filters of LL, taken in the category of bounded distributive lattices; let i a:aMi_a: \uparrow a \to M denote the coproduct inclusion. MM has a prime ideal PP by the PIT for distributive lattices; the pullback P ai a 1(P)P_a \coloneqq i_a^{-1}(P) is a prime ideal of a\uparrow a. Of course P aL{1}P_a \subseteq L \setminus \{1\} by properness of prime ideals.

Put S=L{1}S = L \setminus \{1\}. This is a dcpo by completeness of LL and compactness of 11; since proper ideals ISI \subseteq S are directed subsets, we see that SS admits a sup σ(a)P a\sigma(a) \coloneqq \bigvee P_a for each aSa \in S. We have aσ(a)a \leq \sigma(a) for all aSa \in S. By the Bourbaki-Witt fixed point theorem, the inflationary operator σ:SS\sigma: S \to S has a fixed point, say c:c=σ(c)c: c = \sigma(c). For that cc, note that P ccP_c \subseteq \uparrow c is just {c}\{c\}.

So {c}\{c\} is a prime ideal in c\uparrow c. By meet-irreducibility, this means cc is a prime element in LL.

Remark

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.

Prime elements in quantales of ideals

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 RR be a ring, and let Idl(R)Idl(R) be the sup-lattice of two-sided ideals of RR. Under multiplication of ideals ABA \cdot B, we obtain a quantale structure on Idl(R)Idl(R) whose multiplicative unit is the top element RR. In general we will call a quantale affine (or semicartesian) if its unit is the top element.

The top element 11 of Idl(R)Idl(R) 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 QQ, we say an element pQp \in Q is prime if for all a,bQa, b \in Q we have abpa b \leq p implies apa \leq p or bpb \leq p. Prime elements in Idl(R)Idl(R) are precisely prime ideals of RR.

Theorem

(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 QQ: say aba \equiv b if ( c)1=ac1=bc(\forall_c)\; 1 = a \vee c \Leftrightarrow 1 = b \vee c.

Lemma

The relation \equiv is a quantale congruence.

Proof

First we show that \equiv respects the quantale multiplication \cdot. Suppose aba \equiv b and xyx \equiv y. So for any cc, if 1axc1 \leq a x \vee c, it follows that 1ac1 \leq a \vee c and 1xc1 \leq x \vee c, whence 1bc1 \leq b \vee c and 1yc1 \leq y \vee c. Then

1bc=(b1)c=b(yc)c=bybcc=byc1 \leq b \vee c = (b \cdot 1) \vee c = b(y \vee c) \vee c = b y \vee b c \vee c = b y \vee c

and similarly 1=byc1 = b y \vee c implies 1=axc1 = a x \vee c. So axbya x \equiv b y.

Now we show that \equiv respects joins, i.e., if x iy ix_i \equiv y_i for all iIi \in I, then ix i iy i\bigvee_i x_i \equiv \bigvee_i y_i. Suppose 1=c ix i1 = c \vee \bigvee_i x_i. Since QQ is compact, there is a finite set of the x ix_i, say x 1,,x nx_1, \ldots, x_n, such that 1=cx 1x n1 = c \vee x_1 \vee \ldots \vee x_n. From x ny nx_n \equiv y_n and 1=(cx 1x n1)x n1 = (c \vee x_1 \vee \ldots \vee x_{n-1}) \vee x_n, we derive

1=(cx 1x n1)y n=(y nc)x 1x n11 = (c \vee x_1 \vee \ldots \vee x_{n-1}) \vee y_n = (y_n \vee c) \vee x_1 \vee \ldots \vee x_{n-1}

and we proceed by induction to show 1=cy 1y n1 = c \vee y_1 \vee \ldots \vee y_n, which implies 1c iIy i1 \leq c \vee \bigvee_{i \in I} y_i. Similarly, 1=c iy i1 = c \vee \bigvee_i y_i implies 1=c ix i1 = c \vee \bigvee_i x_i, and so we are done.

Proposition

The quantale Q˜\tilde{Q} formed as the quotient Q/Q/\equiv is a nontrivial compact frame.

Proof

Clearly 1xxc1 \leq x x \vee c implies 1xxcxc1 \leq x x \vee c \leq x \vee c. But also if 1xc1 \leq x \vee c, then

1xc=x1c=x(xc)c=xxxcc=xxc1 \leq x \vee c = x \cdot 1 \vee c = x\cdot (x \vee c) \vee c = x x \vee x c \vee c = x x \vee c

and thus xxxx x \equiv x for all xx. Being an idempotent affine quantale, Q˜\tilde{Q} is a frame.

We have ¬(01)\neg (0 \equiv 1) since (0c=10 \vee c = 1 iff 1c=11 \vee c = 1) fails for c=0c = 0, under the assumption QQ is nontrivial. So Q˜\tilde{Q} is nontrivial. The same proof shows that for any iQi \in Q, we have ¬(i1)\neg (i \equiv 1) if ¬(i=1)\neg (i = 1).

To show Q˜\tilde{Q} is compact, suppose {x i} iI\{x_i\}_{i \in I} is a family in QQ such that ix i1\bigvee_i x_i \equiv 1, in other words such that c ix i=1c \vee \bigvee_i x_i = 1 for all cQc \in Q. Setting c=0c = 0 and applying compactness of QQ, there is a finite subfamily FF such that iFx i=1\bigvee_{i \in F} x_i = 1, and then iFx i1\bigvee_{i \in F} x_i \equiv 1 which proves compactness of Q˜\tilde{Q}.

The quantale quotient map QQ˜Q \to \tilde{Q} preserves arbitrary sups and thus has a right adjoint π:Q˜Q\pi: \tilde{Q} \to Q, by the poset form of the adjoint functor theorem. If we denote the \equiv-class of an element aQa \in Q by [a][a], then for cQ˜c \in \tilde{Q} the element π(c)\pi(c) is given by the explicit formula

π(c)={yQ:[y]c}.\pi(c) = \bigvee \{y \in Q: [y] \leq c\}.
Lemma

Given a regular epi []:QQ˜[-]: Q \to \tilde{Q} in the category of quantales, with right adjoint π:Q˜Q\pi: \tilde{Q} \to Q, if cc is prime in Q˜\tilde{Q}, then π(c)\pi(c) is prime in QQ.

Proof

For elements a,bQa, b \in Q we have

abπ(c) iff [ab]c since[]π iff [a][b]c since[]isaquantalemap iff ([a]c)([b]c) sincecisprime iff (aπ(c))(bπ(c)) since[]π\array{ a b \leq \pi(c) & iff & [a b] \leq c & since \; [-] \dashv \pi \\ & iff & [a] \cdot [b] \leq c & since \; [-]\; is\; \mathrm{a}\; quantale\; map \\ & iff & ([a] \leq c) \vee ([b] \leq c) & since \; c \; is\; prime \\ & iff & (a \leq \pi(c)) \vee (b \leq \pi(c)) & since \; [-] \dashv \pi }

which completes the proof.

The proof of Theorem 3 is immediate:

Proof

Given that QQ is a nontrivial compact affine quantale, Q˜\tilde{Q} is a nontrivial compact frame by Proposition 1. Then Q˜\tilde{Q} has a prime element cc by Banaschewski’s lemma, whence QQ has a prime element by Lemma 2.

Corollary

(PIT for rings) Every unital ring RR has a prime ideal under ZF + (UF).

Remark

See also Banaschewski-Harting. We remark that the nucleus kk on Idl(R)Idl(R) formed by the composition k=π[]:Idl(R)Idl(R)k = \pi \circ [-]: Idl(R) \to Idl(R) has the following properties:

  • k(IJ)=k(IJ)=k(I)k(J)k(I \cap J) = k(I J) = k(I) \cap k(J),

  • k(J)=1k(J) = 1 implies J=1J = 1.

These properties are all that is needed to verify that the fixed points of kk (corresponding to the quotient Idl(R)˜\widetilde{Idl(R)}) 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:

Corollary

Let QQ be a nontrivial compact affine quantale, and let iQi \in Q be any proper element, i.e., assume ¬(i=1)\neg (i = 1). Then there exists a prime element pQp \in Q such that ipi \leq p.

Proof

Let k=π[]:QQk = \pi \circ [-]: Q \to Q be the nucleus corresponding to the congruence \equiv (i.e., the regular epi []:QQ˜[-]: Q \to \tilde{Q}). Since i1i \neq 1, we also have k(i)1k(i) \neq 1 (i.e., ¬(i1)\neg (i \equiv 1)), as shown in the proof of Proposition 1. So Q˜=Fix(k)\tilde{Q} = Fix(k) and the upwards-closed set k(i)Fix(k)k(i) \downarrow Fix(k) are both nontrivial compact affine frames, and there is a prime element pp of k(i)Fix(k)k(i) \downarrow Fix(k). We have a regular epi of quantales rr defined as the composite

Q[]Fix(k)k(i)k(i)Fix(k)Q \stackrel{[-]}{\to} Fix(k) \stackrel{k(i) \vee -}{\to} k(i) \downarrow Fix(k)

(here using the fact that k(i)k(i) \vee - preserves finite meets, since Fix(k)Fix(k) is a distributive lattice). The right adjoint of rr is the inclusion ρ:k(i)Fix(k)Q\rho: k(i) \downarrow Fix(k) \hookrightarrow Q, and it follows from Lemma 2 that p=ρ(p)p = \rho(p) is a prime of QQ such that k(i)pk(i) \leq p. Since ik(i)i \leq k(i), we conclude ipi \leq p.

Generalized prime ideal theorems

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 C\mathbf{C} be a complete, cocomplete biclosed monoidal category (e.g., a cosmos). If AA is a monoid object in C\mathbf{C}, then we can speak of ideals in AA, and typically Idl(A)Idl(A) will inherit good properties from C\mathbf{C}. If for example C\mathbf{C} is regular and well-powered, then Idl(A)Idl(A) forms a quantale under multiplication of ideals, as discussed here.

If we further assume that the monoidal unit II of C\mathbf{C} is compact (i.e., finitely presented), then the top element of Idl(A)Idl(A) generated by the monoid unit e:IAe: I \to A will be a compact element. Then, defining a prime ideal in AA as usual to be a prime element in Idl(A)Idl(A), we deduce existence of prime ideals from Theorem 3.

Here is a sample result along these lines:

Theorem

Let TT be a commutative Lawvere theory. Then the category of algebras Set TSet^T is a well-powered regular cosmos in which the monoidal unit is compact. Consequently, for every monoid AA of Set TSet^T and every proper ideal IAI \hookrightarrow A, there is a prime ideal PP containing II (by applying Corollary 2 to Idl(A)Idl(A)).

Corollary

(PIT for rigs) Every proper ideal in a rig is contained in a prime ideal.

Proof

Take TT to be the theory of commutative monoids, and apply Theorem 4.

References

  • Bernhard Banaschewski and Roswitha Harting, Lattice Aspects of Radical Ideals and Choice Principles, Proc. London Math. Soc. s3-50 (3) (1985), 385-404. (web)
  • Bernhard Banaschewski, Prime Elements from Prime Ideals, Order 2 (1985), 211-213. (Springer link)
  • Jan Paseka, A note on the prime ideal theorem, Acta Universitatis Carolinae, Mathematica et Physica, Vol. 30 (1989), No. 2, 131-136. (pdf)
Revised on February 13, 2016 14:48:59 by Todd Trimble (50.154.86.95)