An algorithmic approach to the existence of ideal objects in commutative algebra

03/07/2019
by   Thomas Powell, et al.
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The existence of ideal objects, such as maximal ideals in nonzero rings, plays a crucial role in commutative algebra. These are typically justified using Zorn's lemma, and thus pose a challenge from a computational point of view. Giving a constructive meaning to ideal objects is a problem which dates back to Hilbert's program, and today is still a central theme in the area of dynamical algebra, which focuses on the elimination of ideal objects via syntactic methods. In this paper, we take an alternative approach based on Kreisel's no counterexample interpretation and sequential algorithms. We first give a computational interpretation to an abstract maximality principle in the countable setting via an intuitive, state based algorithm. We then carry out a concrete case study, in which we give an algorithmic account of the result that in any commutative ring, the intersection of all prime ideals is contained in its nilradical.

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1 Introduction

This paper is an application of proof theory in commutative algebra. To be more precise, we use proof theoretic methods to give a computational interpretation to a general maximality principle (Theorem 2.1), which in particular implies the existence of maximal ideals in commutative rings (Krull’s lemma). In the context of second order arithmetic, the latter statement is equivalent to arithmetical comprehension [41, Chapter III.5], and thus Theorem 2.1 is a genuinely strong principle, and highly non-trivial from a computational perspective.

The extraction of programs from proofs has a long and rich history, dating back to Kreisel’s pioneering work on the ‘unwinding’ of proofs [17, 18]. In the ensuing decades, the application of proof interpretations in particular has become a major topic in proof theory, and today encompasses both proof mining [12, 13, 14], which focuses on obtaining quantitative information primarily from proofs in areas of mathematical analysis, and the mechanized synthesis of programs from proofs, which has found many concrete applications in discrete mathematics and computer science [3, 4, 40].

Though as far back as the 1950s Kreisel already discusses the use of proof theoretic techniques to extract quantitative information from proofs in abstract algebra [19], specifically Hilbert’s 17th problem together with his Nullstellensatz, to date there are comparatively few formal applications of proof interpretations in algebra, the computational analysis of which is done largely on a case by case basis. This typically involves replacing semantic conservation theorems with appropriate syntactic counterparts both sufficient for proofs of elementary statements and provable by elementary means. This method has proved possible in numerous different settings [5, 6, 21, 22, 34, 44], and in the context of commutative algebra the so-called dynamical method is especially dominant [7, 20, 45, 46].111 The second author has contributed to a universal conservation criterion [31, 32, 33] that includes many of the those cases [30, 36, 39]. In dynamical algebra one deals with a supposed ideal object (such as a maximal ideal) only by means of concrete, finitary approximations (such as finitely generated ideals, or rather the finite sets of generators), where the latter provide partial but sufficiently complete information about the former.

Interestingly, the idea of replacing ideal objects with suitable finitary approximations is already implicit in Kreisel’s unwinding program, and is captured by his famous no-counterexample interpretation (n.c.i.). The n.c.i. plays an important role in proof mining, where in particular it corresponds to the notion of metastability [11, 15, 16], which has been made popular by Tao [43] and more recently has featured in higher order computability theory [35].

In this article, we take a new approach to eliminating ideal objects in abstract algebra, by solving an appropriate metastable reformulation of our general maximality principle. We then use this solution to extract direct witnesses from a variant of Krull’s lemma.

The novelty of our approach lies not just in our use of the n.c.i., but in our description of its solution as a state based algorithm, inspired by recent work of the first author [23, 24, 26, 27, 28] which focuses on the algorithmic meaning of extracted programs. This form of presentation allows us to bridge the gap between the rigorous extraction of programs from proofs as terms in some formal calculus, and the more algorithmic style of dynamical algebra.

It also enables us to present our results in an entirely self-contained manner, without needing to introduce any heavy proof theoretic machinery. Though behind the scenes at least, aspects of our work are influenced by Gödel’s functional interpretation [8] and Spector’s bar recursion [42], neither of these make an official appearance, and we have endeavoured to make everything as accessible to the non-specialist as possible.

Our first main contribution, given as Theorems 4.1 and 4.2, is a time sequential algorithm (in the sense of Gurevich [9]), whose states evolve step by step until they terminate in some final state which represents a solution to the n.c.i. of Theorem 2.1. Each step in this process represents an improvement to our construction of an approximate ideal object, and so can also be viewed as a learning procedure in the style of [1].

We then present a concrete application of our abstract result, in which we analyse a classic maximality argument used to prove the well known fact that in any commutative ring, if some element is contained in intersection of all prime ideals, then it must be nilpotent. We show that an instance of our sequential algorithm can be used to directly compute an exponent such that , and thus our case study is another illustration of how the proof theoretic analysis of a highly nonconstructive proof can yield direct, computational information. We conclude by instantiating our algorithm in case of nonconstant coefficients of invertible polynomials. This is a well known example which has been widely studied from a computational perspective [25, 29, 37, 38], thus facilitating a future analysis of our work with other approaches.

2 A general maximality argument

We begin by presenting our abstract maximality principle, which forms the main subject of the paper. Let be some set (which for now is arbitrary but later will be countable), and denote by the set of all finite subsets of . Simple lemmas are stated without proof.

Definition 1

Let be some subset of . We treat as a binary relation and say that the element is generated by the finite set whenever . We extend to arbitrary (not necessarily finite) by defining whenever there exists some finite such that .

Definition 2

Given some , define the sequence of sets by

and let . We say that is the closure of w.r.t. , since whenever then .

Definition 3

For any and , denotes the closed extension of with .

Lemma 1

Suppose that . Then .

Definition 4

Let be some predicate on . For write for . Note in particular that and implies .

Definition 5

We say that is maximal w.r.t. and if

  1. is closed w.r.t. ,

  2. ,

  3. for any .

Theorem 2.1

Suppose that . Then there exists some which is maximal w.r.t. and .

Proof

Define . We show that is nonempty and chain complete w.r.t. set inclusion. Nonemptyness follows from the fact that , so it remains to prove chain completeness. Let be a chain in . Then is clearly closed, and moreover, if then for some , and therefore . This establishes .

Thus by Zorn’s lemma, has some maximal element , which by definition satisfies (i) and (ii). But for we have and thus . But since is closed, it follows that .

Corollary 1

Any commutative ring with has a maximal ideal.

Proof

We follow the standard proof. Define by iff for some and . In addition, define . Then is closed iff it is an ideal, with iff is proper. Now (since ) and if then , thus by Theorem 2.1 there exists some maximal structure . To see that is a maximal ideal, if there were some then we would have for some , and by we would have and thus .

3 A logical analysis of Theorem 2.1

From now on, we assume that is countable and comes equipped with some explicit enumeration . Given some , the initial segment of of length is defined by . Note that . We define by .

Theorem 3.1

Suppose that satisfies

(1)

for all . If then is maximal w.r.t. and .

Proof

Let . We first observe that for all , which follows by induction: For we have and so is true by assumption. Now supposing that holds for some there are two possibilities: If then and hence , and if then and hence . Either way we have .

We now establish each of the maximality conditions in turn. For closure, suppose that but , and so by definition . Since we have for some . First, let . Then and thus , which implies that and thus by Lemma 1

Since this contradicts . But if then and thus implies . But and thus by Lemma 1 again, , contradicting .

That holds is straightforward: For if then and thus follows from . Finally, to show that for , note that implies , and since the result follows.

The purpose of the above theorem was to give a more syntactic formulation of Theorem 2.1 in the countable setting: If then the existence of a some maximal is implied by the existence of some satisfying (1). In order to proceed, we will now take a closer look at the structure of (1) and make some restrictions on the logical complexity of certain parameters.

Lemma 2

Suppose that the relation can be encoded as a -formula. Then the membership relation can also be encoded as a -formula.

Proof

We have iff there exists some finite derivation tree for whose leaves are elements of and whose nodes represent instances of . Given that can be encoded as a -formula, it is clear that the existence of a derivation trees can in turn be represented as -formula via a suitable encoding.

Lemma 3

Suppose that is a -formula and that can be encoded as a -formula. Then is a -formula i.e. for some decidable predicate on .

Proof

We can write for some decidable , and by Lemma 2, for some decidable . Then

and the latter formula can be encoded as for suitable and using some pairing function for the tuple .

Lemma 4

Under the conditions of Lemma 3, (1) holds iff for all :

(2)
Proof

By Lemma 3 setting , so that .

Written out in full, the existence of some satisfying (2) becomes

and so written out in Skolem normal form, this becomes

(3)

This motivates our final version of maximality, which is now in a form where we can directly apply the no-counterexample interpretation.

Definition 6

An explicit maximal object w.r.t. and is a set together with a function such that

for all .

The idea here is that the function provides concrete evidence for why is excluded from the maximal structure : in other words, it encodes an element together with some tree and such that with respect to but fails relative to .

4 An approximating algorithm for maximal objects

In general, it is impossible to effectively compute a set together with an satisfying Definition 6. However, we demonstrate how an approximate, or metastable, formulation of maximality in the spirit of Kreisel’s no-counterexample interpretation, can be directly witnessed via an intuitive stateful procedure.

For a detailed and modern account of the n.c.i., the reader is encouraged to consult e.g. [10, 13]. The rough idea is the following: Given some prenex formula of the form , a functional is said to witness the n.c.i. of if it witnesses i.e. . This definition generalises in the obvious way to prenex formulas of arbitrary complexity. In this section, we give an algorithmic description of such an for being the statement that an explicit maximal object exists, as in Definition 6.

Definition 7

Let be functionals which take as input and and return as output a tuple in . An approximate explicit maximal object w.r.t , and is a set together with a function such that

but now only for and .

Note that Definition 7 is slightly stronger than the n.c.i. (3), since it works for all and not just .

Approximate maximal objects are useful because when a proof of a pure existential statement relies on the existence of some maximal , we are typically able to find functionals which calibrate exactly how this maximal object is used, and thereby construct a witness to the existential statement in terms of an approximate maximal object relative to . We will see an example of this in Section 5.

4.1 The algorithm

We now describe our algorithm which computes approximate maximal objects, as an intuitive state based computation . Here, each is a state, which in this paper is defined to be a function of type i.e. an array , whose th entry is either a natural number or some default value . The idea is that any given state encodes a current approximation to an explicit maximal object: For each state we define the set as

and the function by

where follows from the assumption that . Fixing functionals , we assume for convenience that these now take as input states, and write e.g. for . Define

We now describe how our state evolves. As an initial state, we set

and so and has an empty domain. Now, given that we are in the th state, we carry out the following steps:

  • Search from up to until some is found such that each of the following hold

    • ,

  • If no such is found, the algorithm terminates in state .

  • Otherwise, define

    (where denotes list concatenation) and so in particular, and .

Lemma 5

For all states and we have

Proof

Induction on . For the statement is trivially true, since . So suppose the statement is true for some , and that . Because for some there are two possibilities: either and and so the result follow by the induction hypothesis since and , or and so which is defined to satisfy , and thus the result follows since .

Theorem 4.1

Suppose that the algorithm terminates in state . Then forms an approximate explicit maximal object w.r.t. and .

Proof

If the algorithm terminates, then by definition it holds that for all , if then where . But if then by Lemma 5, and so we’re done.

4.2 Termination

It remains, then, to show that our algorithm actually terminates on some reasonable set of parameters! Here, we make an additional abd completely standard assumption, namely that the functionals are continuous.

Definition 8

We say that are continuous if for all states (which encode ) there exists some such that for any other input state , if then

Note that whenever are instantiated by computable functionals, they will automatically be continuous, so restricting ourselves to the continuous setting is entirely reasonable.

Theorem 4.2

Whenever the algorithm is run on continuous parameters , it terminates after a finite number of steps.

Proof

Suppose that the algorithm does not terminate and thus results in an infinite run . We define a sequence satisfying

(4)

inductively as follows: We let , and if has been defined, either there exists some such that , in which case we define , or for all and we set . To see that this construction satisfies (4) we use induction on . The base case is trivial, so let’s fix some . By the induction hypothesis and the fact that we have for all , and so we only need to check point . Now, in the case for all we’re done since this means that for all . In the other case, if then and observing the manner in which the states evolves at each step, the only way this can change is if is removed from to for some and . But this contradicts the induction hypothesis.

Define to be the limit of the , and let be a point of continuity for on this input. Define

Then in particular, since we must have

But since the algorithm does not terminate, there is some with but . But by definition of , since then implies that for all , a contradiction.

5 Case study: The nilradical as the intersection of all prime ideals

We now use our algorithm to carry out a computational analysis of the following well known fact [2, Proposition 1.8], which is a frequently used form of Krull’s lemma. Recall that a ring element is nilpotent if for some integer .

Theorem 5.1

Let be a countable commutative ring. Suppose that lies in the intersection of all prime ideals of . Then is nilpotent.

We first show how the standard proof follows from our general maximality principle Theorem 2.1.

Proof

Define as in Corollary 1, but now let . Then is closed w.r.t and satisfies iff it is an ideal which does not contain for any . Suppose for contradiction that is not nilpotent, which would mean that and thus hold. By Theorem 2.1 there is some which is maximal w.r.t. and , and in this case is just the ideal generated by and .

Take . Then and hence there exists some such that . Similarly, there exists some with . But then and thus . This would mean that is prime, but then contradicts the assumption that .

Lemma 6

For and defined as in Theorem 5.1, we have

where , as usual denotes the set of lists over and is the length of .

Our aim will be to address the following computational challenge, given any fixed and ,

  • Input. Evidence that lies in the intersection of all prime ideals

  • Output. An exponent such that

The first question is what we take to be evidence that lies in all prime ideals. Note that this assumption is logically equivalent to the statement

so for a computational interpretation of the above it would be reasonable to ask for a procedure which takes some as input, and either confirms that or demonstrates that is not a prime ideal.

Let’s now fix some enumeration of , where we assume for convenience that , and . From now on we assume that we have some function

which for any satisfies

The functional witnesses the statement that or is not a prime ideal.

Lemma 7

Suppose that and satisfy

(5)

where is as in Lemma 6 and if then and . Whenever there exists some nonempty together with a sequence of elements of and such that

Moreover, and are computable in , and .

Proof

This fairly routine case analysis is included in the appendix.

Lemma 8

Suppose that and satisfy (5) as in Lemma 7 and that . Then there exists some , sequence and such that

  • ,

and moreover, , and are computable in , and .

Proof

By Lemma 7 there exist, computable in , and , a nonempty together with and satisfying . In particular, we can find some which is the maximal with , and thus . But by expanding to some sequence with zeroes added wherever needed, we have

where , and thus holds.

Theorem 5.2

Given our input realizer , define the functionals by

Suppose that the algorithm described in Section 4.1 is run on , and for as defined in Lemma 6. Then the algorithm terminates in some final state satisfying

Proof

First of all, we note that are computable, and so in particular must be continuous in the sense of Definition 8. Therefore the algorithm terminates in some final state . By Lemma 5 we have

(6)

We claim that . If this were not the case, then by Lemma 8 and the definition of we would have and for

and so by definition the algorithm cannot be in a final state. This proves the claim. But implies that , and therefore by (6) we have where , which is just

But since we have i.e. .

5.1 Informal description of algorithm

The basic idea behind the algorithm in this section is the following.

  • Each state encodes some , where only if we have found evidence that generates for some , in which case this evidence is encoded as .

  • We start off at with the full set .

  • At state we interact with our functional , which provides us with evidence that either is not a prime ideal, or .

  • If this evidence takes the form of anything other than , then we are able to use this to find some and evidence that generates for some . We exclude from but add all for all (since now the evidence that generates could be falsified by the removal of ).

  • Eventually, using a continuity argument, the algorithm terminates in some state . The only way this can be is if , which indicates that . Thus generates for some encoded in the state.

5.2 Example: Nilpotent coefficients of invertible polynomials

We conclude by outlining a simple and very concrete application [2, pp. 10–11] of Theorem 5.1, and sketching how our algorithm would be implemented in this case. Fixing our countable commutative ring , let be a unit in the polynomial ring . Then each for is nilpotent. To prove this, by Theorem 5.1 it suffices to show that for all prime ideals of .

Let be such that , and let be some arbitrary prime ideal. Then we also have in , but since is prime, is an integral domain, and thus . This implies that in and thus for all .

In order to obtain a concrete algorithm, which for any for , produces some such that , we need to analyse the above argument to produce a specific functional which for any , witnesses the statement that either or is not a prime ideal. Fixing and , we define via the following algorithm:

  • Check in turn whether any of , or are true. In the first case, return , and in the other, and respectively.

  • Otherwise, let be such that for . Then in particular, for we have and so (using that ):

    (7)
    • Either , and since , an analysis of the r.h.s. of (7) allows us to find, in a finite number of steps, either some and such that , in which case we return , or some and such that , in which case we return .

    • Or for some . Take and to be the maximal such that and consider

      Then either , in which case return for or , and since for each summand either or , an analysis identical to the previous case returns for suitable .

Therefore, running our algorithm for as defined above results in a sequential algorithm which, by Theorem 5.2 terminates in some final state with for and .

Example 1

In the very simple case where and , the corresponding run our algorithm for would be as follows;

  • and (since ). Remove with evidence .

  • and (since . Remove with evidence .

  • . Noting that , we have , and so are the maximal with . Then