Algebra in Bishop's style: some major features of the book "A Course in Constructive Algebra'' by Mines, Richman, and Ruitenburg

TL;DR

This paper explores the constructive reinterpretation of classical algebra as presented in 'A Course in Constructive Algebra', highlighting simpler proofs and a clearer understanding achieved without classical logic shortcuts.

Contribution

It analyzes the major features of the book, emphasizing how constructive methods lead to more precise and elegant proofs in algebra.

Findings

Classical theorems are revisited with a constructive perspective.

Proofs become simpler and more elegant without classical logic.

Constructive approach clarifies the true content of algebraic proofs.

Abstract

The book "A Course in Constructive Algebra" (1988) shows the way of understanding classical basic algebra in a constructive style similar to Bishop's Constructive Mathematics. Classical theorems are revisited, with a new flavour, and become much more precise. We are often surprised to find proofs that are simpler and more elegant than the usual ones. In fact, when one cannot use magic tools as the law of excluded middle, it is necessary to understand what is the true content of a classical proof. Also, usual shortcuts allowed in classical proofs introduce sometimes useless detours. In order to understand clearly a problem, prescience may be a handicap.

Equations

$$\mathrm{KernelEmbedding}(\alpha )\in {\mathrm{Hom}}_{\mathbf{A}}(\ker (\alpha ),A)$$

$$u\cdot \mathrm{KernelEmbedding}(\alpha )=\tau ,$$

$$\forall \alpha ,\beta \in {\mathrm{Hom}}_{\mathbf{A}}(A,B):(\alpha =\beta )\vee (\alpha \ne \beta )$$

$$h(b)=\{x\in \{0\}:b=f(a)\text{ for some }a\}.$$

$$I^{n-k}(I^{k}M\cap N)=I^{n}M\cap N.$$

Full text

1. 1 The reception of the book

2. 2 Revisiting Bishop’s set theory

3. 3 The corpus of classical abstract algebra treated in the book

4. 4 Principal ideal domains and finitely generated modules on these rings

5. 5 Factorization problems

6. 6 Noetherian rings, primary decompositions and the principal ideal theorem

7. 7 Wedderburn structure theorem for finite-dimensional $k$-algebras

8. 8 Dedekind domains

9. Acknowledgement

Introduction

The book “A Course in Constructive Algebra” (1988) shows the way of understanding classical basic algebra in a constructive style similar to Bishop’s Constructive Mathematics. Classical theorems are revisited, with a new flavour, and become much more precise. We are often surprised to find proofs that are simpler and more elegant than the usual ones. In fact, when one cannot use magic tools as the law of excluded middle (LEM), it is necessary to understand what is the true content of a classical proof. Also, usual shortcuts allowed in classical proofs introduce sometimes useless detours. In order to understand clearly a problem, prescience may be a handicap.

1 The reception of the book

The reception of the book in France is even more confidential than that of Bishop’s book [2]. I have hardly ever met a French mathematician who has but heard 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 the section “Computer Algebra” of the Journal of Algebra, section whose recommendations to the authors explicitly indicate the interest of the journal for constructive mathematics. What was my surprise when the referee asked me to explain what was the precise meaning of “or” in constructive mathematics, because he was confused and did not understand some arguments. The article was finally rejected in this section of the Journal of Algebra, apparently because of the impossibility of finding a competent referee.

Nevertheless I have recently discovered the following article by Sebastian Posur, A constructive approach to Freyd categories. https://arxiv.org/abs/1712.03492

Here is an excerpt from section 2, “Constructive category theory”. This article seems to me to be a salutary and expected turning point.

To present our algorithmic approach to Freyd categories, we chose the language of constructive mathematics (see, e.g., [MRR88]). We did that for the following reasons: the language of constructive mathematics

reveals the algorithmic content of the theory of Freyd categories, 2. 2.

is perfectly suited for describing generic algorithms, i.e., constructions not depending on particular choices of data structures, 3. 3.

allows us to express our algorithmic ideas without choosing some particular model of computation (like Turing machines) 4. 4.

encompasses classical mathematics, i.e., all results stated in constructive mathematics are also valid classically, 5. 5.

does not differ very much from the classical language in our particular setup.

In constructive mathematics the notions of data types and algorithms (or operations) are taken as primitives and every property must have an algorithmic interpretation. For example given an additive category $\mathbf{A}$ we interpret the property

$\mathbf{A}$ has kernels

as follows: we have algorithms that compute for given

• $\bullet$

$A,B\in\mathrm{Obj}_{\mathbf{A}}$, $\alpha\in\mathrm{Hom}_{\mathbf{A}}(A,B)$, an object $\mathrm{ker}(\alpha)\in\mathrm{Obj}_{\mathbf{A}}$ and a morphism

[TABLE]

for which $\mathrm{KernelEmbedding}(\alpha)\cdot\alpha=0$,

• $\bullet$

$A,B,T\in\mathrm{Obj}_{\mathbf{A}}$, $\alpha\in\mathrm{Hom}_{\mathbf{A}}(A,B)$, $\tau\in\mathrm{Hom}_{\mathbf{A}}(T,A)$ such that $\tau\cdot\alpha=0$ a morphism $u\in\mathrm{Hom}_{\mathbf{A}}(T,\mathrm{ker}(\alpha))$ such that

[TABLE]

where $u$ is uniquely determined (up to $=$) by this property.

Another important example is given by decidable equality, where we interpret the property that for all objects $A,B\in\mathbf{A}$, we have

[TABLE]

as follows: we are given an algorithm that decides or disproves equality of a given pair of morphisms…

On the other hand, we allow ourselves to work classically whenever we interpret Freyd categories in terms of finitely presented functors. The reason for this is pragmatic: we want to demonstrate the usefulness of having Freyd categories computationally available, and we believe that this can be done by interpreting Freyd categories in terms of other categories that classical mathematicians care about.

2 Revisiting Bishop’s set theory

The authors of [CCA] introduce a philosophy of mathematics that differs slightly from that of [2, Bishop, 1967]. This point of view is probably expressed more directly 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 mathematics, but of mathematics open to unpredictable developments, and for which the only criterion of truth is the conviction given by a proof.

The mathematical universe is thus not preexisting, it is on the contrary a properly human construction for the use of the human community.

A novelty is the following. The general point of view is to consider that all mathematics, 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 since they use neither LEM nor Choice. Exactly as the theory of groups is more general 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 construct its members from objects that have been, or could have been, constructed prior 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 conventional: something to be determined when the set is defined, subject only to the requirement that it be an equivalence relation.

A unary relation $P$ on $S$ defines a subset $A=\{x\in S:P(x)\}$ 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 $B$ are subsets of $S$, and if every element of $A$ is an element of $B$, then we say that $A$ is contained in $B$, and write $A\subseteq B$. Two subsets $A$ and $B$ of a set $S$ are equal if $A\subseteq B$ and $B\subseteq A$; this is clearly an equivalence relation on subsets of $S$.

We have described how to construct a subset of $S$, and what it means for two 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 that we 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 example we know how to construct a nonnegative integer, or a real number, or a real function. But this seems problematic since nobody thinks that it is possible to have a universal language for mathematics allowing us to codify these relations. In particular, if the set $\Omega$ of subsets of the singleton $\{0\}$ exists, this means that truth values form a set rather than a class. This seems to say that we know a priori all the truth values that may appear in the future development of mathematics.

In fact, it seems that each time a “set of all subsets of …” is used in the book, this happens in a context where only a well defined set of subsets (in the usual, Bishop, meaning) is necessary. So the set of all subsets is not really needed. Or sometimes the quantification over this set is not needed.111The most important exception is in the definition of well-founded sets and ordinals (see below page 3).

For example let us see the following theorem, whose proof is incredibly simple and elegant.222This theorem is not found in classical textbooks. Bourbaki (Algebra, Chapter VII, paragraph 4, section 1), perhaps the best text for this problem, gives the theorem only for the case $m=n$, $I_{1}\neq R$ and $J_{1}\neq R$. And the proof is less beautiful than in [CCA].

The decomposition in Theorem [V.]2.3 is essentially unique over an arbitrary commutative ring.

[V.]2.4 Theorem. Let $R$ be a commutative ring, $m\leqslant n$ positive integers, and $I_{1}\supseteq I_{2}\supseteq\cdots\supseteq I_{m}$ and $J_{1}\supseteq J_{2}\supseteq\cdots\supseteq J_{n}$ ideals of $R$. Suppose $M$ is an $R$-module that is isomorphic to $\Sigma_{i=1}^{m}R/I_{i}$ and to $\Sigma_{j=1}^{n}R/J_{j}$. Then

1. (a)

$J_{1}=J_{2}=\cdots=J_{n-m}=R$. 2. (b)

$I_{i}=J_{n-m+i}$* for $i=1,\ldots,m$.*

Here there is no hypothesis on the ideals $I_{i}$ and $J_{j}$. If you would want to formalize completely the discourse, you need the quantification over all ideals of $R$, but you don’t really need this complete formalization. Similarly, we do not need 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 in Bishop’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 $\Omega$ of all subsets of $\{0\}$ plays a crucial role. Note also that the nice Theorem I.4.1 seems to be mainly aesthetic, without more concrete applications, 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 $fg=fh$, then $g=h$; that is, $f$ is left cancellable. It is routine to show that $f$ is one-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 and only if it is onto is less routine than the proof of the corresponding result for left cancellable maps.

[I.]4.1 Theorem. A function is right cancellable in the category of sets if and only if it is onto.

Proof.

Suppose $f:A\rightarrow B$ is onto and $gf=hf$. If $b\in B$, 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 $f:A\rightarrow B$ is right cancellable, and let $\Omega$ be the set of all subsets of $\{0\}$. Define $g:B\rightarrow\Omega$ by $g(b)=\{0\}$ for all $b$, and define $h:B\rightarrow\Omega$ by

[TABLE]

Thus $h(b)$ is the subset of $\{0\}$ such that $0\in h(b)$ 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 $\{0\}$. So $g=h$, whence $0\in h(b)$, which means that $b=f(a)$ for some $a$. ∎

In fact, an original feature of [CCA] is the consideration of a notion of category as a fully-fledged mathematical object and not as a simple “manière de parler”:

We deal with two sorts of collections of mathematical objects: sets and categories.

Given two groups, or sets, on the other hand, it is generally incorrect to 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 relation on a set constructs, given any two objects $a$ and $b$ in the set, a proposition ‘$a=b$’. To specify a category $\mathcal{C}$, we must show how to construct, given any two objects $A$ and $B$ in $\mathcal{C}$, a set $\mathcal{C}(A,B)$.

A primary interest of categories is to generalize the notion of a family of objects (indexed by a set). For the category of sets, Bishop [2] considers only families of subsets of a given set. But in usual mathematical practice, and particularly in algebra, we sometimes need a more general notion, which corresponds to the notion of dependent types in the constructive theory of types.

Using the notion of a functor, we can extend our definition of a family of elements of a set to a family of objects in a category $\mathcal{C}$. Let $I$ be a set. A family $A$ of objects of $\mathcal{C}$ indexed by $I$ is a functor from $I$, viewed as a category, to the category $\mathcal{C}$. We often denote such a family by $\{A_{i}\}_{i\in I}$. If $i=j$, then the map from $A_{i}$ to $A_{j}$ is denoted by $A_{j}^{i}$, and is an isomorphism.

With these tools, it is possible to construct important objects in today’s mathematics, as

• •

limits and colimits (e.g. products and coproducts) in some categories,

• •

some algebraic structures freely generated by general sets (not necessarily discrete),

• •

many operations on ordinals (see the definition of ordinals in [CCA] below).

For example, one proves that a module freely generated by a set $S$ is flat; but it is not necessarily projective (Exercise IV.4.9). The classical theorem saying that every module is a quotient of a free module remains valid; the effective consequence is not that the module is a quotient of a projective module, but rather a quotient of a flat module. Thus, by forcing the sets to be discrete (by the aid of LEM), classical mathematics oversimplify the notion of a free module and lead to conclusions impossible to satisfy algorithmically.

A natural notion of ordinal333This notion is different from the ones given by Brouwer or Martin-Löf. See also [5, A constructive theory of ordinals]. is also introduced in chapter I of [CCA], and it is used in classification problems of abelian groups (in chapter XI).

Note that the definition below of a well-founded set uses the quantification over all subsets of $W$.

Let $W$ be a set with a relation $a<b$. A subset $S$ of $W$ is said to be hereditary if $w\in S$ whenever $w^{\prime}\in S$ for each $w^{\prime}<w$. The set $W$ (or the relation $a<b)$ is well founded if each hereditary subset of $W$ equals $W$. A discrete partially ordered set is well founded if the relation $a<b$ (that is, $a\leqslant b$ and $a\neq b)$ on it is well-founded. An ordinal, or a well-ordered set, is a discrete, linearly ordered, well-founded set.

Well-founded sets provide the environment for arguments by induction.

If $\lambda$ and $\mu$ are ordinals, then an injection of $\lambda$ into $\mu$ is a function $\rho$ from $\lambda$ to $\mu$ such that if $a<b$ then $\rho a<\rho b$, and if $c<\rho b$, then there is $a\in\lambda$ such that $\rho a=c$. We shall show that there is at most one injection from $\lambda$ to $\mu.$

[I.]6.5 Theorem. If $\lambda$ and $\mu$ are ordinals, and $\rho$ and $\sigma$ are injections of $\lambda$ into $\mu$, then $\rho=\sigma.$

If there is an injection from the ordinal $\lambda$ to the ordinal $\mu$ we write $\lambda\leqslant\mu$. Clearly compositions of injections are injections, so this relation is transitive. By [Theorem] 6.5 it follows that if $\lambda\leqslant\mu$ and $\mu\leqslant\lambda$, then $\lambda$ and $\mu$ are isomorphic, that is, there is an invertible order preserving function from $\lambda$ to $\mu$. It is natural to say that two isomorphic ordinals are equal.

We are here in a framework close to the constructive theory of dependent types, where all types are created via inductive definitions.

3 The corpus of classical abstract algebra treated in the book

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. Polynomial rings. 6. Matrices and vector spaces. 7. Determinants. 8. Symmetric polynomials. 9. Notes.

Chapter III. Rings and modules.

1. Quasi-regular elements and the Jacobson radical. 2. Coherent and Noetherian modules. 3. Localization. 4. Tensor products. 5. Flat modules. 6. Local rings. 7. Commutative local rings. 8. Notes.

Chapter IV. Divisibility in discrete domains.

1. 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. Finitely presented 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 and transcendence bases. 3. Splitting fields and algebraic closures. 4. Separability and 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) factorial fields. 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-Rees lemma. 3. The Nullstellensatz. 4. Tennenbaum’s approach to the Hilbert basis theorem. 5. Primary ideals. 6. Localization. 7. Primary decompositions. 8. Lasker-Noether rings. 9. Fully Lasker-Noether rings. 10. The principal ideal theorem. 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 subgroups of free groups. 4. Detachable subgroups of finite-rank free groups. 5. Conjugate subgroups. 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 classical theorems to which the constructive reformulation brings a new light and precise additional informations.

We also give some examples of theorems which are trivial in classical mathematics and yet very important from the algorithmic point of view.

4 Principal ideal domains and finitely generated modules on these rings

In classical mathematics, a principal ideal domain is an integral ring in which all ideals are principal. From a constructive point of view, even the two-element field does not satisfy this definition: consider an ideal generated by a binary sequence; finding a generator of this ideal is the same thing as deciding if the sequence is identically zero, which amounts to LPO.

An algorithmically relevant definition, classically equivalent to the classical one, is that of a discrete Bézout integral ring that satisfies a precisely formulated Noetherian condition.

A GCD-monoid is a cancellation [commutative] monoid in which each pair of elements has a greatest common divisor. A GCD-domain is a discrete domain whose nonzero elements form a GCD-monoid.

A principal ideal of a commutative monoid $M$ is a subset $I$ of $M$ such that $I=Ma=\{ma:m\in M\}$ for some $a$ in $M$. We say that $M$ satisfies the divisor chain condition if for each ascending chain $I_{1}\subseteq I_{2}\subseteq I_{3}\subseteq\cdots$ 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 its monoid of nonzero elements does.

[IV.]2.7 Definition. A Bézout domain is a discrete domain such that for 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 the divisor chain condition.

The classical structure theorem says that a finitely generated module on a PID 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 an order where each $a_{i}$ divides the next one.

The purest algorithmic form of this theorem is the theorem of reduction of a 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 equivalent to a matrix in Smith normal form.

[V.]1.4 Theorem. Two $m\times n$ 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 from Theorem V.1.2.

[V.]2.3 Theorem (Structure theorem). Let $M$ be a finitely presented module over a principal ideal domain $R$. Then there exist principal ideals $I_{1}\supseteq I_{2}\supseteq\cdots\supseteq I_{n}$ such that $M$ is isomorphic to the direct sum $R/I_{1}\oplus R/I_{2}\oplus\cdots\oplus R/I_{n}$.

Since the ring is discrete by definition, we can separate the sum into two pieces: the beginning, for indices from $1$ to $k$ say, is the torsion submodule, with $I_{k}=(a_{k})\neq 0$, and the second piece, for $j>k$ with $a_{j}=0$, is a free module of rank $n-k$. On the other hand, in order to know which $I_{j}$’s ($j\leqslant k$) are equal to $R$ (and thus could be removed without damage), we need to have a test of invertibility for elements of $R$, which in this case is equivalent to having a divisibility test between two elements.

In classical mathematics, Theorem V.2.3 is stated for finitely generated modules. From a classical point of view the finitely generated modules over a PID are finitely presented, while from a constructive point of view it is clearly impossible to have an algorithm to achieve this implication, even in the simple case of the $\mathbf{Z}$-module $\mathbf{Z}/I$ where $I$ is countably generated (e.g. generated by a binary sequence).

The way in which Bourbaki (Algebra, chapter VII) treats these theorems deserves to be compared. The structure theorem is given before the Smith reduction theorem for matrices. And the proof, which uses LEM, fails to produce an algorithm to make the theorem explicit.

5 Factorization problems

Theorem IV.4.7 (i) below is usually shown for unique factorization domains, but the 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 prime factors in a GCD monoid satisfying the divisor chain condition is inaccessible from an algorithmic point of view. It is replaced in constructive mathematics by a slightly more subtle theorem. This new theorem can generally be used instead of the classical one when needed to obtain concrete results.

[IV.]1.8 Theorem (Quasi-factorization). Let $x_{1},\ldots,x_{k}$ be elements of a GCD-monoid $M$ satisfying the divisor chain condition. Then there is a family $P$ of pairwise relatively prime elements of $M$ such that each $x_{i}$ is an associate of a product of elements of $P.$

Let $M$ be a cancellation monoid. An element $a\in M$ is said to be bounded by $n$ if whenever $a=a_{0}\cdots a_{n}$ with $a_{i}\in M$, then $a_{i}$ is a unit for some $i$. An element of $M$ is bounded if it is bounded by $n$ for some $n\in\mathbf{N}$; the monoid $M$ is bounded if each of its elements is bounded. A discrete domain 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 versions (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 classical notion.

[IV.]2.1 Definition. A discrete domain $R$ is called a unique factorization domain, or UFD, if each nonzero element $r$ in $R$ is either a unit or has an essentially unique factorization into irreducible elements, that is, if $r=p_{1}\cdots p_{m}$ and $r=q_{1}\cdots q_{n}$ are two factorizations of $r$ into irreducible elements, then $m=n$ and we can reindex so that $p_{i}\sim q_{i}$ for each $i$. We say 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 the classical notion, but they introduce algorithmically relevant distinctions, totally invisible in classical mathematics, due to the use of LEM, which annihilates these relevant distinctions. In Theorem IV.4.7 the points (ii) (attached to the point (i)) and (vi) (i.e. (i) and (v)) are two distinct, inequivalent versions of the same classical theorem about UFDs.

[IV.]4.7 Theorem. Let $R$ be a discrete domain.

1. (i)

If $R$ is a GCD-domain, then so is $R[X].$ 2. (ii)

If $R$ is bounded, then so is $R[X].$ 3. (iii)

If $R$ has recognizable units, then so does $R[X].$ 4. (iv)

If $R$ has decidable divisibility, then so does $R[X].$ 5. (v)

If $R$ satisfies the divisor chain condition, then so does $R[X].$ 6. (vi)

If $R$ is a quasi-UFD, then so is $R[X].$

Concerning factorization problems for polynomials over a discrete field, the algorithmic 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 trivial thing, contrarily to what is stated in classical mathematics.

Chapter VII of [CCA] explores the factorization problems in polynomial rings in great detail.

The basic constructive theorem on this subject is given in Chapter VI. As it happens that the characteristic of a field or a ring is not known in advance, but can be revealed during a construction, some precautions are necessary in the statements, 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 the alternative “$k$ has a nonzero nonunit”. But it happens in [CCA] that the theorem is used in the precise form given here, e.g. in Chapter IX about the structure of finite-dimensional algebras.

[VI.]6.3 Theorem. Let $k$ be a discrete commutative ring with recognizable units, and $S$ a finite set of monic polynomials in $k[X]$. Then either $k$ has a nonzero nonunit or we can construct a finite set $T$ of monic polynomials in $k[X]$ such that

1. (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 $k$. 2. (ii)

Distinct elements of $T$ are strongly relatively prime. 3. (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 of univariate polynomials, a family of separable strongly relatively prime monic polynomials which gives a more precise version of the quasi-factorization theorem IV.1.8 (which deals with quasi-UFDs).

6 Noetherian rings, primary decompositions and the principal ideal theorem

An $R$-module is said to be strongly discrete if finitely generated submodules are detachable.444In [CCA], the terminology is “module with detachable submodules”, it was later replaced by “strongly discrete module”. See e.g. [12, Richman 1998]. It is said to be coherent555Bourbaki (Algebra, Chapter X, or Commutative Algebra Chapter I) calls pseudo coherent 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 be linked 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 coherent modules. if any finitely generated submodule is finitely presented. The notion of strongly discrete coherent ring is fundamental from the algorithmic point of view in commutative algebra. In particular for the following reason: on a strongly discrete coherent ring, linear systems are perfectly understood and mastered.666In the article of Posur cited above, these rings are called “computable”.

In usual textbooks in classical mathematics, this notion is usually hidden behind that of a Noetherian ring, and rarely put forward. In classical mathematics every Noetherian ring $R$ is coherent because every submodule of $R^{n}$ is finitely generated, and every finitely generated module is coherent for the same reason. 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 a satisfying constructive formulation of Noetherianity which implies coherence, and coherence is often the most important property from an algorithmic point of view. Consequently, from a constructive point of view, coherence must be added when we use the notion of a Noetherian ring or module.

The definition adopted for Noetherian module in [CCA] is: a module in which any ascending chain of finitely generated submodules admits two equal consecutive terms. It is a constructively acceptable definition, equivalent in classical mathematics to the usual definition.

The classical theorem stating that over a Noetherian ring every finitely generated $A$-module is Noetherian is often advantageously replaced by the following constructive 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 Noetherian coherent.

Two important classical results about Noetherian rings have constructive proofs within the framework given by [CCA].

[VIII.]2.7 Theorem (Artin-Rees). Let $I$ be a finitely generated ideal of a coherent commutative Noetherian ring $R$. Let $N$ be a finitely generated submodule of a finitely presented $R$-module $M$. Then there is $k$ such that for all $n\geqslant k$ we have

[TABLE]

[VIII.]2.8 Theorem (Krull intersection theorem). Let $M$ be a finitely presented module over a coherent commutative Noetherian ring $R$, and let $I$ be a finitely generated ideal of $R$. Let $A=\cap_{n}I^{n}M$. Then $a\in Ia$ for each $a\in A$, so $IA=A$.

Hilbert basis theorem

Which are the coherent rings $R$ such that the polynomial rings $R[X_{1},\ldots,X_{n}]$ are also coherent? From a constructive point of view, we know two classes of rings sharing this property: coherent Noetherian rings (see below) and Prüfer domains (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] for polynomial rings over a discrete field). These proofs are very clearly laid out in [CCA].

[VIII.]1.5 Theorem (Hilbert basis theorem). If $R$ is a coherent Noetherian ring, then so is $R[X]$. If, in addition, $R$ has detachable left ideals,777i.e., is left strongly discrete then so does $R[X]$.

There is an analogous theorem in Computer Algebra (see [1, 1994, Theorem 4.2.8]) saying that for a coherent Noetherian strongly discrete ring $R$, there is a “Gröbner basis algorithm” computing the leading ideal of a finitely generated ideal in $R[X_{1},\ldots,X_{n}]$ for a given monomial order. In fact, this Computer Algebra theorem and Theorem VIII.1.5 are essentially the same result. One is easily deduced from the other.

Nevertheless we note that algorithms for these theorems are quite different from each other. Moreover, authors in 1994 seem to ignore that the problem was solved 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 as depending on the data.)

Primary decomposition theorem

[CCA] gives an adequate constructive theory of primary decompositions. This is based on the work of Seidenberg, [16, 1978] and [17, 1984]. In [CCA] this work is made more simple and synthetic.

Let $R$ be a commutative ring. An ideal $Q$ of $R$ is said to be primary if $xy\in Q$ implies $x\in Q$ or $y^{n}\in Q$ for some $n$. One sees that $\sqrt{Q}$ is a prime ideal $P$.

[CCA] gives a variant w.r.t. the usual terminology, with no importance in the case of Noetherian rings for classical mathematics: ideals are all finitely generated. A primary decomposition of an ideal $I$ in a commutative ring is a finite family of finitely generated primary ideals $Q_{1},\ldots,Q_{n}$ such that the $\sqrt{Q_{i}}$ are finitely generated and $I=\bigcap_{i}Q_{i}$. In this case the ideal $I$ is said to be decomposable. In classical mathematics, every ideal of a Noetherian ring has a primary decomposition.

In a constructive framewok, which convenient hypotheses do we have to add for a coherent Noetherian strongly discrete ring in order to get primary decompositions? A possible answer is the following one, given in [CCA].

A Lasker-Noether ring is a coherent Noetherian ring with detachable ideals such that the radical of each finitely generated ideal is the intersection of a finite number of finitely generated prime ideals.

This definition is constructively acceptable and applies to usual examples like $\mathbf{Z}$, $\mathbf{Q}[X]$, and $k[X]$ when $k$ is an algebraically closed discrete field: they are clearly constructively Lasker-Noether for this definition. Many other usual examples are also available, as explained below.

In fact, when $k$ is a discrete field, $k[X]$ is easily seen to be Lasker-Noether if and only if $k$ is a factorial field. This equivalence has no meaning in classical mathematics since all fields are factorial. Nevertheless it should be possible to state an analogous result for mechanical computations using Turing machines.

The first properties of Lasker-Noether rings are summarized in three theorems.

[VIII.]8.1 Theorem. Let $S$ be a multiplicative submonoid of a Lasker-Noether ring $R$ such that $I\cap S$ is either empty or nonempty for each finitely generated ideal $I$ of $R$. Then $S^{-1}R$ is a Lasker-Noether ring.

If $S=R\setminus P$ for a prime ideal $P$, condition “$I\cap S$ 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 that for 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 a finitely generated ideal of $R$. Then $R/I$ is a Lasker-Noether ring.

[VIII.]8.5 Theorem (Primary decomposition theorem). Let $R$ be a Lasker-Noether ring. Then each finitely generated ideal of $R$ has a primary decomposition.

Principal ideal theorem

A more elaborate property of Lasker-Noether rings is the famous principal ideal theorem of Krull and the fact that finitely generated proper prime ideals have a well-defined height.

[VIII.]10.4 Theorem (Generalized principal ideal theorem). Let $R$ be a Lasker-Noether ring. Let $I=(a_{1},\ldots,a_{n})$. Then every minimal prime ideal over $I$ has height at most $n$.

[VIII.]10.5 Theorem. Let $P$ be a finitely generated proper prime ideal of a 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: which convenient hypotheses do we have to add for a Lasker-Noether ring $R$ in order to get that $R[X_{1},\ldots,X_{n}]$ is also Lasker-Noether?

Call $R$ a fully Lasker-Noether ring if it is a Lasker-Noether ring and if for each finitely generated prime ideal $P$ of $R$, the field of quotients of $R/P$ is fully factorial. Note that the ring of integers $\mathbf{Z}$ is a fully Lasker-Noether ring, 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 acceptable definition equivalent to the definition of a Noetherian ring in classical mathematics), a very large number of classical theorems concerning Noetherian rings now 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 is constructively stronger than the one in [CCA]. The usual examples of Noetherian rings are Noetherian in this meaning. With this notion, the definition of a Lasker-Noether ring becomes more natural: it is a Noetherian coherent strongly discrete ring in which we have a primality test for finitely generated ideals. The paper gives a nice theory of fully Lasker-Noether rings in this context.

Note. The computation of primary decompositions in polynomial rings over discrete fields or over $\mathbf{Z}$ is an active area of research in Computer Algebra. The seminal paper of Seidenberg is sometimes cited, but not the book [CCA].

7 Wedderburn structure theorem for finite-dimensional $k$-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 isomorphic to a finitely generated subalgebra of an algebra of matrices $E_{k}(k^{n})$ (the algebra of $k$-endomorphisms of the vector space $k^{n}$). We shorten the terminology by speaking 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 $1+xA\subseteq A^{\times}$. It is a (two-sided) ideal and the Jacobson radical of the quotient $A/I$ is zero.

When $A$ is a $k$-algebra of finite dimension, this radical can also be defined as the nilpotent radical: $\mathop{\mathrm{rad}}\nolimits(A)$ 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 $xa_{1}\cdots xa_{n}$ is zero.

Let $A$ be a $k$-algebra of finite dimension. We can construct a basis of the center 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 ideal generated by a finite part of $A$. But it may be difficult to construct a basis of the radical, and we cannot generally state that the radical is finite-dimensional (over $k$).

Nevertheless, we know how to construct objects whose counterparts are trivial 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 theorem.

[IX.]3.3 Theorem. Let $A$ be a finite-dimensional $k$-algebra and $L$ a finite-dimensional (left) ideal of $A$. Then either $L\cap\mathop{\mathrm{rad}}\nolimits A\neq 0$ or $A=L\oplus N$ for some (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 the algebra is discrete (as in the present context) the definition amounts to saying that 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. Here is the constructive reformulation given in [CCA]. A field $k$ is called separably factorial when separable polynomials in $k[X]$ have a prime decomposition.

We now characterize separably factorial fields in terms of decomposing algebras into products of simple algebras. This is the first part of Wedderburn’s theorem.

[IX.]4.3 Theorem. A discrete field $k$ is separably factorial if and only if every finite-dimensional $k$-algebra with zero radical is a product of simple algebras.

A clarification concerning the ability to construct a basis of the radical is given in the following corollary.

[IX.]4.5 Corollary. A discrete field $k$ is fully factorial if and only if every finite-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 algebras says that a finite-dimensional simple algebra is isomorphic to a full ring of matrices over a division algebra.

The constructive version of this theorem given in [CCA] elucidates in a surprising way the computational content of this classical theorem.

[IX.]5.1 Theorem. Let $A$ be a finite-dimensional $k$-algebra, and $L$ a nontrivial left ideal of $A$. Then either

1. (i)

$A$* has a nonzero radical* 2. (ii)

$A$* is a product of finite-dimensional $k$-algebras* 3. (iii)

$A$* is isomorphic to a full matrix ring over some $k$-algebra of dimension less than $A$.*

The fundamental problem is to be able to recognize whether a given finite-dimensional algebra is a division algebra or not, in the sense of being able either to assert that it is a division algebra or to construct a nontrivial left ideal. If we could do that, then Theorem IX.5.1 would imply that every finite-dimensional $k$-algebra has a finite dimensional radical, and modulo its radical it is a product of full matrix rings over division algebras. This condition 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 $k$ are equivalent.

1. (i)

Each finite-dimensional $k$-algebra is either a division algebra or has a nontrivial left ideal. 2. (ii)

Each finite-dimensional left module $M$ over a finite dimensional $k$-algebra $A$ is either reducible or irreducible. 3. (iii)

Each finite-dimensional $k$-algebra $A$ has a finite-dimensional radical, and $A/\mathop{\mathrm{rad}}\nolimits A$ is a product of full matrix rings over division algebras.

And we remain a little disappointed with these questions at the end of chapter IX.

For what fields $k$ do the conditions of Theorem 5.2 hold? Finite fields and algebraically closed fields provide trivial examples. The field of algebraic real numbers admits only three finite-dimensional division algebras, and a constructive proof of this statement shows that this field satisfies the conditions of Theorem 5.2.

[IX.]5.3 Theorem. Let $k$ be a discrete subfield of $\mathbf{R}$ that is algebraically closed in $\mathbf{R}$, and $H=k(i,j)$ the quaternion algebra over $k$. If $A$ is a finite-dimensional algebra over $k$, then either $A$ has a zero-divisor, or $A$ is isomorphic to $k$, to $k(i)$, or to $H$.

Does the field $\mathbf{Q}$ of rational numbers satisfy the conditions of Theorem 5.2? Certainly we are not going to produce a Brouwerian counterexample when $k=\mathbf{Q}$. Probably a close analysis of the classical theory of division algebras over $\mathbf{Q}$, in analogy with Theorem 5.3, will yield a proof.

8 Dedekind domains

Although it is commonly felt that algebraic number theory is essentially constructive in its classical form, even those authors who pay particular attention to the constructive aspects of the theory employ highly nonconstructive techniques which nullify their efforts. In [3, Borevich-Shafarevich, 1966], for example, it is assumed that every polynomial can be factored into a product of irreducible polynomials (every field is factorial) and that given a nonempty subset of the positive integers you can find its least element.

The constructive theory of Dedekind domains in [CCA] allows us to give an explicit version of the classical statements of number theory and algebraic geometry concerning local fields, for example in the book of J.-P. Serre [18]. This theory also gives the appropriate hypotheses to account for the classical results concerning Dedekind domains, as found, for example, in Bourbaki.

This requires giving sufficiently precise and binding definitions, beginning with 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 discrete valuations on a Heyting field $k$ is a Dedekind set if

1. (i)

For each $x\in k$ there is a finite subset $T$ of $S$ so that $|x|_{p}\leqslant 1$ for each $p\in S\setminus T$. 2. (ii)

If $q$ and $q^{\prime}$ are distinct valuations of $S$, and $\varepsilon>0$, then there exists $x\in k$ with $|x|_{p}\leqslant 1$ for each $p\in S$, such that $|x-1|_{q}<\varepsilon$ and $|x|_{q^{\prime}}<\varepsilon$. Hence distinct valuations are inequivalent.

Let $S$ be a Dedekind set of valuations on a Heyting field $k$. If $p\in S$, then, because $p$ is nonarchimedean, the set $R(p)=\{x\in k:|x|_{p}\leqslant 1\}$ is a ring, which is local as $p$ is discrete. We call $R(p)$ the local ring at $p$. The elements of the ring $\bigcap_{p\in S}R(p)$ are called the integers at $S$. A ring is a Dedekind domain if it is the ring of integers at a Dedekind set of valuations on a Heyting field.

If the strong point is to give a constructive account of most of the classical theorems, a weak point is that for example a PID is a Dedekind domain only in the case where we have algorithms of factorization of principal ideals into prime ideals.

We can compare this for example with the exposition in [6], where a definition is given that is constructively weaker but closer to the usual classical definition (see Definition XII-7.7 and Theorem XII-7.9). In [6], Dedekind domains have quasi-factorization of finite sets of finitely generated ideals, and the total factorization Dedekind domains correspond to the discrete Dedekind domains of [CCA].

Acknowledgement

I am indebted to Thierry Coquand and Stefan Neuwirth for many relevant comments and suggestions.