Algebra in Bishop's style: some major features of the book "A Course in Constructive Algebra'' by Mines, Richman, and Ruitenburg
AAlgebra in Bishop’s style:some major features of the book“ A Course in Constructive Algebra”by Mines, Richman, and Ruitenburg
Henri LombardiMarch 12, 2019
Abstract
The book “A Course in Constructive Algebra” (1988) shows the wayof understanding classical basic algebra in a constructive style similarto Bishop’s Constructive Mathematics. Classical theorems are revisited,with a new flavour, and become much more precise. We are often surprisedto find proofs that are simpler and more elegant than the usual ones. Infact, when one cannot use magic tools as the law of excluded middle, it isnecessary to understand what is the true content of a classical proof. Also,usual shortcuts allowed in classical proofs introduce sometimes uselessdetours. In order to understand clearly a problem, prescience may be ahandicap.
Contents k -algebras 168 Dedekind domains 18Références 20 a r X i v : . [ m a t h . HO ] M a r ntroduction The book “A Course in Constructive Algebra” (1988) shows the way of under-standing classical basic algebra in a constructive style similar to Bishop’s Con-structive Mathematics. Classical theorems are revisited, with a new flavour,and become much more precise. We are often surprised to find proofs that aresimpler and more elegant than the usual ones. In fact, when one cannot usemagic tools as the law of excluded middle (LEM), it is necessary to understandwhat is the true content of a classical proof. Also, usual shortcuts allowed inclassical proofs introduce sometimes useless detours. In order to understandclearly a problem, prescience may be a handicap.
The reception of the book in France is even more confidential than that ofBishop’s book [2]. I have hardly ever met a French mathematician who has butheard of the existence of the book.The Computer Algebra community could be expected to be a little more up-to-date since all theorems in [CCA] have a computational content, and could,at least in principle, be implemented in the usual Computer Algebra softwares.Some years ago I have submitted an article of constructive algebra to thesection “Computer Algebra” of the Journal of Algebra, section whose recom-mendations to the authors explicitly indicate the interest of the journal forconstructive mathematics. What was my surprise when the referee asked meto explain what was the precise meaning of “or” in constructive mathematics,because he was confused and did not understand some arguments. The articlewas finally rejected in this section of the Journal of Algebra, apparently becauseof the impossibility of finding a competent referee.Nevertheless I have recently discovered the following article by SebastianPosur,
A constructive approach to Freyd categories . https://arxiv.org/abs/1712.03492 Here is an excerpt from section 2, “Constructive category theory”. This articleseems to me to be a salutary and expected turning point.To present our algorithmic approach to Freyd categories, we chose thelanguage of constructive mathematics (see, e.g., [MRR88]). We did that forthe following reasons: the language of constructive mathematics1. reveals the algorithmic content of the theory of Freyd categories,2. is perfectly suited for describing generic algorithms, i.e., constructionsnot depending on particular choices of data structures,3. allows us to express our algorithmic ideas without choosing some par-ticular model of computation (like Turing machines)4. encompasses classical mathematics, i.e., all results stated in construc-tive mathematics are also valid classically,2. does not differ very much from the classical language in our particularsetup.In constructive mathematics the notions of data types and algorithms(or operations) are taken as primitives and every property must have analgorithmic interpretation. For example given an additive category A weinterpret the property A has kernelsas follows: we have algorithms that compute for given • A, B ∈ Obj A , α ∈ Hom A ( A, B ) , an object ker( α ) ∈ Obj A and amorphism KernelEmbedding( α ) ∈ Hom A (ker( α ) , A ) for which KernelEmbedding( α ) · α = 0 , • A, B, T ∈ Obj A , α ∈ Hom A ( A, B ) , τ ∈ Hom A ( T, A ) such that τ · α =0 a morphism u ∈ Hom A ( T, ker( α )) such that u · KernelEmbedding( α ) = τ, where u is uniquely determined (up to = ) by this property.Another important example is given by decidable equality , where weinterpret the property that for all objects A, B ∈ A , we have ∀ α, β ∈ Hom A ( A, B ) : ( α = β ) ∨ ( α (cid:54) = β ) as follows: we are given an algorithm that decides or disproves equality ofa given pair of morphisms. . .On the other hand, we allow ourselves to work classically whenever weinterpret Freyd categories in terms of finitely presented functors. The reasonfor this is pragmatic: we want to demonstrate the usefulness of having Freydcategories computationally available, and we believe that this can be doneby interpreting Freyd categories in terms of other categories that classicalmathematicians care about. The authors of [CCA] introduce a philosophy of mathematics that differs slightlyfrom that of [2, Bishop, 1967]. This point of view is probably expressed moredirectly in the papers [10, 11] and in the book [4].First of all, as in Bishop, the point of view is not that of formalized mathe-matics, but of mathematics open to unpredictable developments, and for whichthe only criterion of truth is the conviction given by a proof.The mathematical universe is thus not preexisting, it is on the contrary aproperly human construction for the use of the human community.A novelty is the following. The general point of view is to consider that all3athematics, classical as well as constructive, deal with the same ideal objects.The unique difference is in the tools used for the investigation of this universe.Constructive mathematics are more general than classical mathematics sincethey use neither LEM nor Choice. Exactly as the theory of groups is moregeneral than the theory of abelian groups, since commutativity is not assumed.Let us quote a passage.Our notion of what constitutes a set is a rather liberal one. [I.]2.1 Definition.
A set S is defined when we describe how to constructits members from objects that have been, or could have been, constructedprior to S , and describe what it means for two members of S to be equal.Following Bishop we regard the equality relation on a set as conven-tional: something to be determined when the set is defined, subject only tothe requirement that it be an equivalence relation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A unary relation P on S defines a subset A = { x ∈ S : P ( x ) } of S : an element of A is an element of S that satisfies P , and two elements of A are equal if and only if they are equal as elements of S . If A and B aresubsets of S , and if every element of A is an element of B , then we say that A is contained in B , and write A ⊆ B . Two subsets A and B of a set S are equal if A ⊆ B and B ⊆ A ; this is clearly an equivalence relation onsubsets of S .We have described how to construct a subset of S , and what it means fortwo subsets of S to be equal. Thus we have defined the set of all subsets,or the power set , of S .This is rather surprising for a follower of Bishop. The authors of [CCA]think that the notion of “a unary relation defined on a given set” is so clear thatwe may consider a well-defined set of all these unary relations. In other words,we know how to construct these unary relations, in a similar way as for examplewe know how to construct a nonnegative integer, or a real number, or a realfunction. But this seems problematic since nobody thinks that it is possible tohave a universal language for mathematics allowing us to codify these relations.In particular, if the set Ω of subsets of the singleton { } exists, this meansthat truth values form a set rather than a class. This seems to say that weknow a priori all the truth values that may appear in the future development ofmathematics.In fact, it seems that each time a “set of all subsets of . . . ” is used in thebook, this happens in a context where only a well defined set of subsets (inthe usual, Bishop, meaning) is necessary. So the set of all subsets is not reallyneeded. Or sometimes the quantification over this set is not needed. For example let us see the following theorem, whose proof is incredibly simpleand elegant. The most important exception is in the definition of well-founded sets and ordinals (seebelow page 6). This theorem is not found in classical textbooks. Bourbaki (Algebra, Chapter VII, para-graph 4, section 1), perhaps the best text for this problem, gives the theorem only for the case m = n , I (cid:54) = R and J (cid:54) = R . And the proof is less beautiful than in [CCA]. [V.]2.4 Theorem. Let R be a commutative ring, m (cid:54) n positive integers,and I ⊇ I ⊇ · · · ⊇ I m and J ⊇ J ⊇ · · · ⊇ J n ideals of R . Suppose M isan R -module that is isomorphic to Σ mi =1 R/I i and to Σ nj =1 R/J j . Then(a) J = J = · · · = J n − m = R .(b) I i = J n − m + i for i = 1 , . . . , m . Here there is no hypothesis on the ideals I i and J j . If you would want toformalize completely the discourse, you need the quantification over all ideals of R , but you don’t really need this complete formalization. Similarly, we do notneed to quantify over the class of all commutative rings when we write: “Let R be a commutative ring”. See [8, Dependent sums and dependent products inBishop’s set theory] for a formal system using class quantification.Note however the following passage which deals with the category of sets,and where the set Ω of all subsets of { } plays a crucial role. Note also thatthe nice Theorem I.4.1 seems to be mainly aesthetic, without more concreteapplications, within the framework of the theory of the categories.[...] The categorical property corresponding to a function f being one-to-one is that if g and h are maps from any set C to A , and f g = f h ,then g = h ; that is, f is left cancellable . It is routine to show that f isone-to-one if and only if it is left cancellable.A map f from A to B is onto if for each b in B there exists a in A such that f ( a ) = b . The corresponding categorical property is that f be right cancellable , that is, if g and h are maps from B to any set C , and gf = hf , then g = h . The proof that a function f is right cancellable if andonly if it is onto is less routine than the proof of the corresponding resultfor left cancellable maps. [I.]4.1 Theorem. A function is right cancellable in the category of sets ifand only if it is onto.Proof.
Suppose f : A → B is onto and gf = hf . If b ∈ B , then there exists a in A such that f ( a ) = b . Thus g ( b ) = g ( f ( a )) = h ( f ( a )) = h ( b ) , so g = h .Conversely suppose f : A → B is right cancellable, and let Ω be the set ofall subsets of { } . Define g : B → Ω by g ( b ) = { } for all b , and define h : B → Ω by h ( b ) = { x ∈ { } : b = f ( a ) for some a } . Thus h ( b ) is the subset of { } such that ∈ h ( b ) if and only if there exists a such that b = f ( a ) . Clearly gf = hf is the map that takes every element of A to the subset { } . So g = h , whence ∈ h ( b ) , which means that b = f ( a ) for some a . 5n fact, an original feature of [CCA] is the consideration of a notion ofcategory as a fully-fledged mathematical object and not as a simple “manièrede parler”:We deal with two sorts of collections of mathematical objects: sets andcategories.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Given two groups, or sets, on the other hand, it is generally incorrectto ask if they are equal; the proper question is whether or not they are isomorphic , or, more generally, what are the homomorphisms between them.A category, like a set, is a collection of objects. An equality relationon a set constructs, given any two objects a and b in the set, a proposition ‘ a = b ’. To specify a category C , we must show how to construct, given anytwo objects A and B in C , a set C ( A, B ) .A primary interest of categories is to generalize the notion of a family ofobjects (indexed by a set). For the category of sets, Bishop [2] considers onlyfamilies of subsets of a given set. But in usual mathematical practice, and partic-ularly in algebra, we sometimes need a more general notion, which correspondsto the notion of dependent types in the constructive theory of types.Using the notion of a functor, we can extend our definition of a familyof elements of a set to a family of objects in a category C . Let I be a set. A family A of objects of C indexed by I is a functor from I , viewed as acategory, to the category C . We often denote such a family by { A i } i ∈ I . If i = j , then the map from A i to A j is denoted by A ij , and is an isomorphism.With these tools, it is possible to construct important objects in today’smathematics, as • limits and colimits (e.g. products and coproducts) in some categories, • some algebraic structures freely generated by general sets (not necessarilydiscrete), • many operations on ordinals (see the definition of ordinals in [CCA] be-low).For example, one proves that a module freely generated by a set S is flat; butit is not necessarily projective (Exercise IV.4.9). The classical theorem sayingthat every module is a quotient of a free module remains valid; the effectiveconsequence is not that the module is a quotient of a projective module, butrather a quotient of a flat module. Thus, by forcing the sets to be discrete (bythe aid of LEM), classical mathematics oversimplify the notion of a free moduleand lead to conclusions impossible to satisfy algorithmically.A natural notion of ordinal is also introduced in chapter I of [CCA], and itis used in classification problems of abelian groups (in chapter XI). This notion is different from the ones given by Brouwer or Martin-Löf. See also [5, Aconstructive theory of ordinals]. W .Let W be a set with a relation a < b . A subset S of W is said to be hereditary if w ∈ S whenever w (cid:48) ∈ S for each w (cid:48) < w . The set W (or therelation a < b ) is well founded if each hereditary subset of W equals W .A discrete partially ordered set is well founded if the relation a < b (thatis, a (cid:54) b and a (cid:54) = b ) on it is well-founded. An ordinal , or a well-orderedset , is a discrete, linearly ordered, well-founded set.Well-founded sets provide the environment for arguments by induction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .If λ and µ are ordinals, then an injection of λ into µ is a function ρ from λ to µ such that if a < b then ρa < ρb , and if c < ρb , then there is a ∈ λ such that ρa = c . We shall show that there is at most one injectionfrom λ to µ. [I.]6.5 Theorem. If λ and µ are ordinals, and ρ and σ are injections of λ into µ , then ρ = σ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .If there is an injection from the ordinal λ to the ordinal µ we write λ (cid:54) µ . Clearly compositions of injections are injections, so this relation istransitive. By [Theorem] 6.5 it follows that if λ (cid:54) µ and µ (cid:54) λ , then λ and µ are isomorphic, that is, there is an invertible order preserving functionfrom λ to µ . It is natural to say that two isomorphic ordinals are equal .We are here in a framework close to the constructive theory of dependenttypes, where all types are created via inductive definitions. Basic classical algebra is fairly widely covered by the various chapters of [CCA].Perhaps the best is to recall the table of contents of the book.Chapter I. Sets.1. Constructive vs. classical mathematics. 2. Sets, subsets and functions.3. Choice. 4. Categories. 5. Partially ordered sets and lattices. 6. Well-founded sets and ordinals. 7. Notes.Chapter II. Basic algebra.1. Groups. 2. Rings and fields. 3. Real numbers. 4. Modules. 5. Polyno-mial rings. 6. Matrices and vector spaces. 7. Determinants. 8. Symmetricpolynomials. 9. Notes.Chapter III. Rings and modules.1. Quasi-regular elements and the Jacobson radical. 2. Coherent and Noethe-rian modules. 3. Localization. 4. Tensor products. 5. Flat modules. 6. Localrings. 7. Commutative local rings. 8. Notes.Chapter IV. Divisibility in discrete domains.7. Divisibility in cancellation monoids. 2. UFD’s and Bézout domains. 3. Dedekind-Hasse rings and Euclidean domains. 4. Polynomial rings. 5. Notes.Chapter V. Principal ideal domains. 1. Diagonalizing matrices. 2. Finitelypresented modules. 3. Torsion modules, p -components, elementary divisors. 4.Linear transformations. 5. Notes.Chapter VI. Field theory.1. Integral extensions and impotent rings. 2. Algebraic independence andtranscendence bases. 3. Splitting fields and algebraic closures. 4. Separabilityand diagonalizability. 5. Primitive elements. 6. Separability and characteristic p . 7. Perfect fields. 8. Galois theory. 9. Notes.Chapter VII. Factoring polynomials.1. Factorial and separably factorial fields. 2. Extensions of (separably) factorialfields. 3. Seidenberg fields. 4. The fundamental theorem of algebra. 5. Notes.Chapter VIII. Commutative Noetherian rings.1. The Hilbert basis theorem. 2. Noether normalization and the Artin-Reeslemma. 3. The Nullstellensatz. 4. Tennenbaum’s approach to the Hilbert basistheorem. 5. Primary ideals. 6. Localization. 7. Primary decompositions.8. Lasker-Noether rings. 9. Fully Lasker-Noether rings. 10. The principal idealtheorem. 11. Notes.Chapter IX. Finite dimensional algebras.1. Representations. 2. The density theorem. 3. The radical and summands.4. Wedderburn’s theorem, part one. 5. Matrix rings and division algebras.6. Notes.Chapter X. Free groups.1. Existence and uniqueness. 2. Nielsen sets. 3. Finitely generated subgroupsof free groups. 4. Detachable subgroups of finite-rank free groups. 5. Conjugatesubgroups. 6. Notes.Chapter XI. Abelian groups.1. Finite-rank torsion-free groups. 2. Divisible groups. 3. Height functions on p -groups. 4. Ulm’s theorem. 5. Construction of Ulm groups. 6. Notes.Chapter XII. Valuation theory.1. Valuations. 2. Locally precompact valuations. 3. Pseudofactorial fields.4. Normed vector spaces. 5. Real and complex fields. 6. Hensel’s lemma.7. Extensions of valuations. 8. e and f . 9. Notes.Chapter XIII. Dedekind domains.1. Dedekind sets of valuations. 2. Ideal theory. 3. Finite extensions.Bibliography. Index.In the following sections we comment some significant examples of classicaltheorems to which the constructive reformulation brings a new light and preciseadditional informations.We also give some examples of theorems which are trivial in classical math-ematics and yet very important from the algorithmic point of view.8 Principal ideal domains and finitely generatedmodules on these rings
In classical mathematics, a principal ideal domain is an integral ring in whichall ideals are principal. From a constructive point of view, even the two-elementfield does not satisfy this definition: consider an ideal generated by a binarysequence; finding a generator of this ideal is the same thing as deciding if thesequence is identically zero, which amounts to LPO.An algorithmically relevant definition, classically equivalent to the classicalone, is that of a discrete Bézout integral ring that satisfies a precisely formulatedNoetherian condition.A
GCD-monoid is a cancellation [commutative] monoid in which eachpair of elements has a greatest common divisor. A
GCD-domain is adiscrete domain whose nonzero elements form a GCD-monoid.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A principal ideal of a commutative monoid M is a subset I of M suchthat I = M a = { ma : m ∈ M } for some a in M . We say that M satisfiesthe divisor chain condition if for each ascending chain I ⊆ I ⊆ I ⊆ · · · of principal ideals, there is n such that I n = I n +1 .A discrete domain is said to satisfy the divisor chain condition if itsmonoid of nonzero elements does. [IV.]2.7 Definition. A Bézout domain is a discrete domain such thatfor each pair of elements a, b there is a pair s, t such that sa + tb divides a and b . A principal ideal domain is a Bézout domain which satisfies thedivisor chain condition.The classical structure theorem says that a finitely generated module on aPID is a direct sum of a finite rank free submodule and of the torsion submodule,itself equal to a direct sum of modules R/ ( a i ) with the non-zero a i put in anorder where each a i divides the next one.The purest algorithmic form of this theorem is the theorem of reduction ofa matrix into a Smith normal form.A matrix A = ( a ij ) is in Smith normal form if it is diagonal and a ii | a i +1 ,i +1 for each i. [V.]1.2 Theorem. Each matrix over a principal ideal domain is equivalentto a matrix in Smith normal form. [V.]1.4 Theorem.
Two m × n matrices in Smith normal form over a GCD-domain are equivalent if and only if corresponding elements are associates. The structure theorem for finitely presented modules follows directly fromTheorem V.1.2. 9
V.]2.3 Theorem (Structure theorem).
Let M be a finitely presentedmodule over a principal ideal domain R . Then there exist principal ideals I ⊇ I ⊇ · · · ⊇ I n such that M is isomorphic to the direct sum R/I ⊕ R/I ⊕ · · · ⊕ R/I n . Since the ring is discrete by definition, we can separate the sum into twopieces: the beginning, for indices from to k say, is the torsion submodule,with I k = ( a k ) (cid:54) = 0 , and the second piece, for j > k with a j = 0 , is a freemodule of rank n − k . On the other hand, in order to know which I j ’s ( j (cid:54) k )are equal to R (and thus could be removed without damage), we need to have atest of invertibility for elements of R , which in this case is equivalent to havinga divisibility test between two elements.In classical mathematics, Theorem V.2.3 is stated for finitely generated mod-ules. From a classical point of view the finitely generated modules over a PIDare finitely presented, while from a constructive point of view it is clearly im-possible to have an algorithm to achieve this implication, even in the simplecase of the Z -module Z /I where I is countably generated (e.g. generated by abinary sequence).The way in which Bourbaki (Algebra, chapter VII) treats these theoremsdeserves to be compared. The structure theorem is given before the Smithreduction theorem for matrices. And the proof, which uses LEM, fails to producean algorithm to make the theorem explicit. Theorem IV.4.7 (i) below is usually shown for unique factorization domains, butthe underlying Noetherian condition is in fact useless. [IV.]4.7 Theorem.
Let R be a discrete domain. (i) If R is a GCD-domain, then so is R [ X ] . The reader is invited to appreciate the elegance of the proof in [CCA].The classical theorem of factorization of an element into a product of primefactors in a GCD monoid satisfying the divisor chain condition is inaccessiblefrom an algorithmic point of view. It is replaced in constructive mathematicsby a slightly more subtle theorem. This new theorem can generally be usedinstead of the classical one when needed to obtain concrete results. [IV.]1.8 Theorem (Quasi-factorization).
Let x , . . . , x k be elements ofa GCD-monoid M satisfying the divisor chain condition. Then there is afamily P of pairwise relatively prime elements of M such that each x i is anassociate of a product of elements of P. M be a cancellation monoid. An element a ∈ M is said to be bounded by n if whenever a = a · · · a n with a i ∈ M , then a i is a unit forsome i . An element of M is bounded if it is bounded by n for some n ∈ N ;the monoid M is bounded if each of its elements is bounded. A discretedomain is bounded if its nonzero elements form a bounded monoid.A GCD-domain satisfying the divisor chain condition is called a quasi-UFD .The quasi-UFDs and the bounded GCD-domains are two constructive ver-sions (that are not constructively equivalent) of the classical notion of a UFD.In fact, we find in [CCA] still three other constructive versions of this classicalnotion. [IV.]2.1 Definition. A discrete domain R is called a unique factoriza-tion domain , or UFD , if each nonzero element r in R is either a unit orhas an essentially unique factorization into irreducible elements, that is, if r = p · · · p m and r = q · · · q n are two factorizations of r into irreducibleelements, then m = n and we can reindex so that p i ∼ q i for each i . Wesay that R is factorial if R [ X ] is a UFD.Call a discrete field k fully factorial if any finite-dimensional extension of k is factorial.The five constructive versions are in classical mathematics equivalent to theclassical notion, but they introduce algorithmically relevant distinctions, totallyinvisible in classical mathematics, due to the use of LEM, which annihilatesthese relevant distinctions. In Theorem IV.4.7 the points (ii) (attached to thepoint (i)) and (vi) (i.e. (i) and (v)) are two distinct, inequivalent versions of thesame classical theorem about UFDs. [IV.]4.7 Theorem. Let R be a discrete domain.(i) If R is a GCD-domain, then so is R [ X ] . (ii) If R is bounded, then so is R [ X ] . (iii) If R has recognizable units, then so does R [ X ] . (iv) If R has decidable divisibility, then so does R [ X ] . (v) If R satisfies the divisor chain condition, then so does R [ X ] . (vi) If R is a quasi-UFD, then so is R [ X ] . Concerning factorization problems for polynomials over a discrete field, thealgorithmic situation is not correctly described by classical mathematics. E.g.factorization of polynomials in k [ X ] where k is a discrete field is not a trivialthing, contrarily to what is stated in classical mathematics.11hapter VII of [CCA] explores the factorization problems in polynomialrings in great detail.The basic constructive theorem on this subject is given in Chapter VI. As ithappens that the characteristic of a field or a ring is not known in advance, butcan be revealed during a construction, some precautions are necessary in thestatements, as below in point (i). Note that if we discover a prime number p equal to zero in a ring k , it is necessarily unique (unless the ring is trivial).In the following theorem, if k is a discrete field, then we simply drop thealternative “ k has a nonzero nonunit”. But it happens in [CCA] that the theoremis used in the precise form given here, e.g. in Chapter IX about the structure offinite-dimensional algebras. [VI.]6.3 Theorem. Let k be a discrete commutative ring with recognizableunits, and S a finite set of monic polynomials in k [ X ] . Then either k hasa nonzero nonunit or we can construct a finite set T of monic polynomialsin k [ X ] such that(i) Each element of T is of the form f ( X q ) where f is separable, and q = 1 or q is a power of a prime that is zero in k .(ii) Distinct elements of T are strongly relatively prime.(iii) Every polynomial in S is a product of polynomials in T . When k is a discrete field, we thus obtain, starting from a given family ofunivariate polynomials, a family of separable strongly relatively prime monicpolynomials which gives a more precise version of the quasi-factorization theo-rem IV.1.8 (which deals with quasi-UFDs). An R -module is said to be strongly discrete if finitely generated submodulesare detachable. It is said to be coherent if any finitely generated submodule isfinitely presented. The notion of strongly discrete coherent ring is fundamentalfrom the algorithmic point of view in commutative algebra. In particular forthe following reason: on a strongly discrete coherent ring, linear systems areperfectly understood and mastered. In usual textbooks in classical mathematics, this notion is usually hiddenbehind that of a
Noetherian ring, and rarely put forward. In classical mathe- In [CCA], the terminology is “module with detachable submodules”, it was later replacedby “strongly discrete module”. See e.g. [12, Richman 1998]. Bourbaki (Algebra, Chapter X, or Commutative Algebra Chapter I) calls pseudo coher-ent module what [CCA] calls coherent module (as in quasi all texts in english literature),and coherent module what [CCA] calls finitely presented coherent module. This is to belinked to “Faisceaux Algébriques Cohérents” by J.-P. Serre. Note also that the Stacks Project(Collective work, http://stacks.math.columbia.edu ) uses Bourbaki’s definition for coherentmodules. In the article of Posur cited above, these rings are called “computable”. R is coherent because every submodule of R n isfinitely generated, and every finitely generated module is coherent for the samereason. Furthermore, we have the Hilbert basis theorem, which states that if R is Noetherian, then every finitely presented R -algebra is also a Noetherian ring, whereas the same statement does not hold if one replaces “Noetherian” with“coherent” (see [19, Soublin, 1970]).From an algorithmic point of view however, it seems impossible to find asatisfying constructive formulation of Noetherianity which implies coherence,and coherence is often the most important property from an algorithmic pointof view. Consequently, from a constructive point of view, coherence must beadded when we use the notion of a Noetherian ring or module.The definition adopted for Noetherian module in [CCA] is: a module inwhich any ascending chain of finitely generated submodules admits two equalconsecutive terms. It is a constructively acceptable definition, equivalent inclassical mathematics to the usual definition.The classical theorem stating that over a Noetherian ring every finitely gen-erated A -module is Noetherian is often advantageously replaced by the followingconstructive theorems. Over a coherent ring (resp. strongly discrete coherent) every finitely presented A -module is coherent (resp. strongly discrete coherent) . Over a Noetherian coherent ring every finitely presented A -module is Noetheriancoherent .Two important classical results about Noetherian rings have constructiveproofs within the framework given by [CCA]. [VIII.]2.7 Theorem (Artin-Rees). Let I be a finitely generated ideal ofa coherent commutative Noetherian ring R . Let N be a finitely generatedsubmodule of a finitely presented R -module M . Then there is k such thatfor all n (cid:62) k we have I n − k ( I k M ∩ N ) = I n M ∩ N. [VIII.]2.8 Theorem (Krull intersection theorem). Let M be a finitelypresented module over a coherent commutative Noetherian ring R , and let I be a finitely generated ideal of R . Let A = ∩ n I n M . Then a ∈ Ia for each a ∈ A , so IA = A . Hilbert basis theorem
Which are the coherent rings R such that the polynomial rings R [ X , . . . , X n ] are also coherent? From a constructive point of view, we know two classes ofrings sharing this property: coherent Noetherian rings (see below) and Prüferdomains (see [20, Yengui, 2015, Chapter 4]).The Hilbert basis theorem for the definition of Noetherianity given in [CCA]is Theorem VIII.1.5 below. Proofs go back to 1974 ([9, Richman, 1974] and [15,Seidenberg, 1974], see also [13, Seidenberg, 1971] and [14, Seidenberg, 1973] forpolynomial rings over a discrete field). These proofs are very clearly laid out in[CCA]. 13 VIII.]1.5 Theorem (Hilbert basis theorem). If R is a coherentNoetherian ring, then so is R [ X ] . If, in addition, R has detachable leftideals, then so does R [ X ] . There is an analogous theorem in Computer Algebra (see [1, 1994, Theorem4.2.8]) saying that for a coherent Noetherian strongly discrete ring R , there isa “Gröbner basis algorithm” computing the leading ideal of a finitely generatedideal in R [ X , . . . , X n ] for a given monomial order. In fact, this ComputerAlgebra theorem and Theorem VIII.1.5 are essentially the same result. One iseasily deduced from the other.Nevertheless we note that algorithms for these theorems are quite differentfrom each other. Moreover, authors in 1994 seem to ignore that the problem wassolved essentially in 1974, and algorithms in [1] are not certified constructively(in fact, from the proof, no bound can be estimated for the number of steps asdepending on the data.) Primary decomposition theorem [CCA] gives an adequate constructive theory of primary decompositions. Thisis based on the work of Seidenberg, [16, 1978] and [17, 1984]. In [CCA] thiswork is made more simple and synthetic.Let R be a commutative ring. An ideal Q of R is said to be primary if xy ∈ Q implies x ∈ Q or y n ∈ Q for some n . One sees that √ Q is a primeideal P .[CCA] gives a variant w.r.t. the usual terminology, with no importance inthe case of Noetherian rings for classical mathematics: ideals are all finitelygenerated. A primary decomposition of an ideal I in a commutative ring is afinite family of finitely generated primary ideals Q , . . . , Q n such that the √ Q i are finitely generated and I = (cid:84) i Q i . In this case the ideal I is said to be decomposable . In classical mathematics, every ideal of a Noetherian ring hasa primary decomposition.In a constructive framewok, which convenient hypotheses do we have toadd for a coherent Noetherian strongly discrete ring in order to get primarydecompositions? A possible answer is the following one, given in [CCA].A Lasker-Noether ring is a coherent Noetherian ring with detachableideals such that the radical of each finitely generated ideal is the intersectionof a finite number of finitely generated prime ideals.This definition is constructively acceptable and applies to usual exampleslike Z , Q [ X ] , and k [ X ] when k is an algebraically closed discrete field: theyare clearly constructively Lasker-Noether for this definition. Many other usualexamples are also available, as explained below.In fact, when k is a discrete field, k [ X ] is easily seen to be Lasker-Noetherif and only if k is a factorial field. This equivalence has no meaning in classicalmathematics since all fields are factorial. Nevertheless it should be possible tostate an analogous result for mechanical computations using Turing machines.14he first properties of Lasker-Noether rings are summarized in three theo-rems. [VIII.]8.1 Theorem. Let S be a multiplicative submonoid of a Lasker-Noether ring R such that I ∩ S is either empty or nonempty for each finitelygenerated ideal I of R . Then S − R is a Lasker-Noether ring. If S = R \ P for a prime ideal P , condition “ I ∩ S is either empty or nonempty”means that “ I either is contained in P or is not”. Since I is finitely generated,the test exists if and only if P is detachable. So, theorem VIII.8.1 implies thatfor each detachable prime ideal, and so for each finitely generated prime ideal,the localization R P is Lasker-Noether. [VIII.]8.2 Theorem. Let R be a Lasker-Noether ring, and let I be afinitely generated ideal of R . Then R/I is a Lasker-Noether ring. [VIII.]8.5 Theorem (Primary decomposition theorem).
Let R be aLasker-Noether ring. Then each finitely generated ideal of R has a primarydecomposition. Principal ideal theorem
A more elaborate property of Lasker-Noether rings is the famous principal idealtheorem of Krull and the fact that finitely generated proper prime ideals havea well-defined height. [VIII.]10.4 Theorem (Generalized principal ideal theorem).
Let R be a Lasker-Noether ring. Let I = ( a , . . . , a n ) . Then every minimal primeideal over I has height at most n . [VIII.]10.5 Theorem. Let P be a finitely generated proper prime ideal ofa Lasker-Noether ring R . Then there is m such that P has height m , and P is a minimal prime over some ideal generated by m elements. Fully Lasker-Noether rings
Finally, it is important to give a constructive answer to the following: whichconvenient hypotheses do we have to add for a Lasker-Noether ring R in orderto get that R [ X , . . . , X n ] is also Lasker-Noether?Call R a fully Lasker-Noether ring if it is a Lasker-Noether ring and iffor each finitely generated prime ideal P of R , the field of quotients of R/P is fully factorial. Note that the ring of integers Z is a fully Lasker-Noetherring, as is any fully factorial field.The following three theorems (with the previous theorems about Lasker-Noether rings) show that in this context (i.e. with this constructively acceptable15efinition equivalent to the definition of a Noetherian ring in classical mathe-matics), a very large number of classical theorems concerning Noetherian ringsnow have a constructive proof and a clear meaning. It sounds like a “miracle”of the same kind as Bishop’s book. [VIII.]9.1 Theorem. Let I be a finitely generated ideal of a fully Lasker-Noether ring R . Then R/I is a fully Lasker-Noether ring. [VIII.]9.2 Theorem. If P is a detachable prime ideal of a fully Lasker-Noether ring R , then R P is a fully Lasker-Noether ring. [VIII.]9.6 Theorem. If R is a fully Lasker-Noether ring, then so is R [ X ] .Note. The paper [7, Perdry, 2004] defines a notion of Noetherianity which isconstructively stronger than the one in [CCA]. The usual examples of Noetherianrings are Noetherian in this meaning. With this notion, the definition of aLasker-Noether ring becomes more natural: it is a Noetherian coherent stronglydiscrete ring in which we have a primality test for finitely generated ideals. Thepaper gives a nice theory of fully Lasker-Noether rings in this context.
Note.
The computation of primary decompositions in polynomial rings overdiscrete fields or over Z is an active area of research in Computer Algebra. Theseminal paper of Seidenberg is sometimes cited, but not the book [CCA]. k -algebras We deal here with unitary associative k -algebras which are finite-dimensional k -vector spaces on a discrete field k . In other words, these algebras are isomorphicto a finitely generated subalgebra of an algebra of matrices E k ( k n ) (the algebraof k -endomorphisms of the vector space k n ). We shorten the terminology byspeaking of “ k -algebra of finite dimension”.If A is a not necessarily commutative ring, its Jacobson radical is the set I of elements x such that xA ⊆ A × . It is a (two-sided) ideal and the Jacobsonradical of the quotient A/I is zero.When A is a k -algebra of finite dimension, this radical can also be defined asthe nilpotent radical : rad( A ) is the set of elements x such that the left ideal xA is nilpotent, i.e. there exists an integer n such that every product xa · · · xa n is zero.Let A be a k -algebra of finite dimension. We can construct a basis of thecenter of A as well as the minimal polynomial over k of an arbitrary element of A .We can also construct a basis of the left ideal and another of the two-sided idealgenerated by a finite part of A . But it may be difficult to construct a basis ofthe radical, and we cannot generally state that the radical is finite-dimensional(over k ).Nevertheless, we know how to construct objects whose counterparts are triv-ial in classical mathematics (if we do not try to construct them!). For example,as an alternative to the construction of the radical, we have the following theo-rem. 16 IX.]3.3 Theorem.
Let A be a finite-dimensional k -algebra and L a finite-dimensional (left) ideal of A . Then either L ∩ rad A (cid:54) = 0 or A = L ⊕ N forsome (left) ideal N . A module M is reducible if it has a nontrivial submodule—otherwise it is irreducible (or simple ).A k -algebra is said to be simple if each two-sided ideal is trivial. When thealgebra is discrete (as in the present context) the definition amounts to sayingthat if an element is nonzero, the (two-sided) ideal it generates contains 1.The first part of Wedderburn’s structure theorem says that every finite-dimensional k -algebra with zero radical is a product of simple algebras. Hereis the constructive reformulation given in [CCA]. A field k is called separablyfactorial when separable polynomials in k [ X ] have a prime decomposition.We now characterize separably factorial fields in terms of decomposingalgebras into products of simple algebras. This is the first part of Wedder-burn’s theorem. [IX.]4.3 Theorem. A discrete field k is separably factorial if and only ifevery finite-dimensional k -algebra with zero radical is a product of simplealgebras. A clarification concerning the ability to construct a basis of the radical isgiven in the following corollary. [IX.]4.5 Corollary.
A discrete field k is fully factorial if and only if everyfinite-dimensional algebra A over k has a finite-dimensional nilpotent ideal I such that A/I is a product of simple k -algebras. The second part of Wedderburn’s structure theorem for semi-simple algebrassays that a finite-dimensional simple algebra is isomorphic to a full ring ofmatrices over a division algebra.The constructive version of this theorem given in [CCA] elucidates in asurprising way the computational content of this classical theorem. [IX.]5.1 Theorem.
Let A be a finite-dimensional k -algebra, and L a non-trivial left ideal of A . Then either(i) A has a nonzero radical(ii) A is a product of finite-dimensional k -algebras(iii) A is isomorphic to a full matrix ring over some k -algebra of dimensionless than A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17he fundamental problem is to be able to recognize whether a givenfinite-dimensional algebra is a division algebra or not, in the sense of beingable either to assert that it is a division algebra or to construct a nontrivialleft ideal. If we could do that, then Theorem IX.5.1 would imply that everyfinite-dimensional k -algebra has a finite dimensional radical, and moduloits radical it is a product of full matrix rings over division algebras. Thiscondition is equivalent to being able to recognize whether an arbitrary finite-dimensional representation of a finite-dimensional k -algebra is reducible. [IX.]5.2 Theorem. The following conditions on a discrete field k areequivalent.(i) Each finite-dimensional k -algebra is either a division algebra or has anontrivial left ideal.(ii) Each finite-dimensional left module M over a finite dimensional k -algebra A is either reducible or irreducible.(iii) Each finite-dimensional k -algebra A has a finite-dimensional radical,and A/ rad A is a product of full matrix rings over division algebras. And we remain a little disappointed with these questions at the end of chap-ter IX.For what fields k do the conditions of Theorem 5.2 hold? Finite fieldsand algebraically closed fields provide trivial examples. The field of alge-braic real numbers admits only three finite-dimensional division algebras,and a constructive proof of this statement shows that this field satisfies theconditions of Theorem 5.2. [IX.]5.3 Theorem. Let k be a discrete subfield of R that is algebraicallyclosed in R , and H = k ( i, j ) the quaternion algebra over k . If A is afinite-dimensional algebra over k , then either A has a zero-divisor, or A isisomorphic to k , to k ( i ) , or to H . Does the field Q of rational numbers satisfy the conditions of Theorem5.2? Certainly we are not going to produce a Brouwerian counterexamplewhen k = Q . Probably a close analysis of the classical theory of divisionalgebras over Q , in analogy with Theorem 5.3, will yield a proof. Although it is commonly felt that algebraic number theory is essentiallyconstructive in its classical form, even those authors who pay particularattention to the constructive aspects of the theory employ highly noncon-structive techniques which nullify their efforts. In [3, Borevich-Shafarevich,1966], for example, it is assumed that every polynomial can be factored into18 product of irreducible polynomials (every field is factorial) and that givena nonempty subset of the positive integers you can find its least element.The constructive theory of Dedekind domains in [CCA] allows us to give anexplicit version of the classical statements of number theory and algebraic ge-ometry concerning local fields, for example in the book of J.-P. Serre [18]. Thistheory also gives the appropriate hypotheses to account for the classical resultsconcerning Dedekind domains, as found, for example, in Bourbaki.This requires giving sufficiently precise and binding definitions, beginningwith those in the theory of (rank-one) valuations.For example, let us see the definitions concerning Dedekind domains. [XIII.]1.1 Definition.
A nonempty discrete set S of nontrivial discretevaluations on a Heyting field k is a Dedekind set if(i) For each x ∈ k there is a finite subset T of S so that | x | p (cid:54) for each p ∈ S \ T .(ii) If q and q (cid:48) are distinct valuations of S , and ε > , then there exists x ∈ k with | x | p (cid:54) for each p ∈ S , such that | x − | q < ε and | x | q (cid:48) < ε .Hence distinct valuations are inequivalent.Let S be a Dedekind set of valuations on a Heyting field k . If p ∈ S ,then, because p is nonarchimedean, the set R ( p ) = { x ∈ k : | x | p (cid:54) } isa ring, which is local as p is discrete. We call R ( p ) the local ring at p .The elements of the ring (cid:84) p ∈ S R ( p ) are called the integers at S . A ringis a Dedekind domain if it is the ring of integers at a Dedekind set ofvaluations on a Heyting field.If the strong point is to give a constructive account of most of the classicaltheorems, a weak point is that for example a PID is a Dedekind domain only inthe case where we have algorithms of factorization of principal ideals into primeideals.We can compare this for example with the exposition in [6], where a def-inition is given that is constructively weaker but closer to the usual classicaldefinition (see Definition XII-7.7 and Theorem XII-7.9). In [6], Dedekind do-mains have quasi-factorization of finite sets of finitely generated ideals, andthe total factorization Dedekind domains correspond to the discrete Dedekinddomains of [CCA].
Acknowledgement
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