mathematics of the seventies
In this edition:

Computable Analysis: R.L. Goodstein
Prime Number Theory: H. Halberstam
Simple Noetherian Rings: R. Hart

Computable Analysis
R Goodstein

A computable (recursive) function from N, the class of natural numbers, into N is, roughly speaking, one whose value for any given argument is determined by an ideal computer with unlimited memory (a Turing machine).

The notion is readily generalized to the field of rational numbers R by means of a standard enumeration of the nationals.

A computably convergent sequence is a computable function f:N --> R with associated modulus of convergence, a computable function c:N --> N, such that n >= c(r) ==> | f(n) - f(c(r))| < 2^-r.

A real number a is computable if there is a computably convergent sequence an such that a = limit as n tends to infinity an . We shall take for granted the extension of these ideas to computably convergent complex sequences, and the natural definitions of computable continuity. The computable complex numbers form a field, and the roots of a polynomial with computable complex coefficients are computable complex numbers.

It is well known from classical analysis that the coefficients of a polynomial are continuous functions of the roots and that the roots vary continuously with the coefficients. Recently E. Specker formulated and proved the analogues of this result in computable analysis.

Since two polynomials have the same roots if and only if their coefficients are proportional, a polynomial (of degree m) may be represented as a point in complex projective space P^m . Let the subset of P^m with rational coordinates be denoted by Pmr, and choose a metric on Pmr (compatible with its topology) such that the distance | u - v | between points u, v has rational values on Pmr x Pmr and is computable on this product space.

The space of root-systems of polynomials of degree at most m is the symmetric product T^rn of m two spheres S^2; a point in T^m has coordinates c1, ... , cm, each ci being a point of S^2, and two sets (c1, ... , cm), (c1, ... , cm) represent the same point of T^m if there is a permutalion pi such that ci = cpi(i), for all i, 1 <= i <= m..

Let Tmr be the subset of T^m of points with complex rational coordinates, and choose a metric on T^m (compatible with its topology so that the distance | u - v | is rational and computable on Tmr x Tmr. The map f from T^m into P^m which associates a polynomial with its system of roots, maps Tmr computably into Pmr and is computably continuous. The fundamental theory of algebra, in classical analysis, assures us that f is a map of T^m onto P^m, and since f is injective and T^m is compact, the inverse f^-1 is continuous.

The corresponding result in computable analysis is that f^-1 is a computably continuous and computable topological function on P^m relative to Pmr.

This result can be extended to general computable metric spaces which are totally bounded, this generalization being the constructive analogue of the theorem asserting the continuity of the inverse function of a continuous injective function defined on a compact space.

Prime Number Theory
H Halberstam

The prime numbers are notoriously slow in yielding up their secrets. There are many results which, while the experimental and heuristic evidence in their favour is overwhelming, have no prospect of early proof in view and, indeed, lie deeper than the famous Riemann hypothesis. Take Bouniakovsky's conjecture (1857): according to this, if f is an irreducible polynomial with integer coefficients (and.positive leading coefficient), without fixed prime divisors (n^2 + n + 2 is always divisible by 2), then f(n) is prime for infinitely many integer values of n.

Writing k for the degree of f, the only value of k for which this conjecture is known to be true is k = 1; this follows from Dirichlet's famous result (1837) that the arithmetic progression an+b (n = 0, 1, 2, ... ; (a,b)=1) contains infinitely many primes

For k = 2 we have already an open problem - it is not known whether n^2 +1 is prime infinitely often (i.o.). The best we can say at the present time is that n^2 +1 = P3 i.o., where Pr is an almost prime of order r, that is, a number having at most r prime factors (equal or distinct); and this is a special case of the general result that, for each f of the above kind and of degree k, f(n) = Pk+1 i.o.

This result has been claimed by various people, notably Barban and Buchstab in Russia, but the fullest published account is to be found in a paper by H. E. Richert, Mathematika, 1969.

We may formulate a related conjecture according to which f(p) is prime i.o. (i.e. for infinitely many primes p) - here f is as in Bouniakovsky's conjecture, f(n)=n and f satisfies one other necessary condition; in this case there is the approximate result that f(p) = P2k+1 i.o. - for example p^2 + p + 1 = P5 i.o; the last mentioned result is a contribution to the famous 'twin prime' conjecture, namely that p + 2 = p' i.o.

The same approach yields also the best available approximation to Goldbach's conjecture (actually first raised by Descartes) that 2n = p+p' for all n > 2 - we can show now that 2n = p+P3, for all sufficiently large n (see Richert loc.cit. for a pretty full account).

The prime twin conjecture is difficult enough, but this does not prevent us from raising the question of prime triplets (e.g. the infinitude of triplets of primes p, p+2, p+6), 'quadruplets', etc. and indeed L.E.Dickson conjectured in 1904 (and I'm sure others did before him) that there exist infinitely many instances of each of these also. Recently J.Porter, a student of mine, showed, for example, that (p+2)(p+6) = P7 i.o. as an approximation to the prime triplet problem, and similar results, too complicated to state here in their full generality now exist for products of linear, and other, polynomials, with integer or prime arguments.

To cite two illustrations, the product of three linear polynomials is, infinitely often a P10; and if the argument is constrained to be a prime, it is, i.o. a P14. If one takes polynomials in more than one variable one expects, intuitively, to do better (see the article by Cassels, MANIFOLD-9, on the existence of a prime generating polynomial in 1000 variables); and, indeed, G.Greaves (J. of Number Theory 1971) has shown recently, as part of a more general result, that an irreducible homogeneous cubic polynomiial in two variables (without fixed prime divisors) is i.o. a P2.

All these results have been achieved by an extension of the famous sieve of Eratosthenes. The method is combinatorial in principle, but relies on substantial aids from analysis and requires also deep information about the way primes are distributed in arithmetic progressions. The same method yields also analogues for almost-primes of other famous problems in prime number theory, such as those relating to gaps between consecutive primes. and the least prime in an arithmetic progression.

Thus, although all these classical problems about primes are still far from settled, we do now have approximate results which are sufficiently close to be of some independent interest.

Simple Noetherian Rings
R Hart

For the purposes of this note, rings are associative with unit element, and 'Noetherian' means that the maximum condition holds for left and right ideals. The topic of Noetherian rings is a big one; I have chosen simple rings to facilitate a brief account, without too many definitions.

It is useful to observe first of all that a ring with minimum condition (left and right) is Noetherian. A famous theorem of Artin-Wedderburn says that every simple ring with minimum condition is isomorphic to a complete matrix ring over a division. (The converse is also true, and easy to prove). Some examples of simple Noetherian rings without minimum condition are given later. A theorem of Goldie says that every simple Noetherian ring has a ring of quotients which is a simple ring with minimum condition. This tells us where to look.for simple Nottherian, but on the other hand there are a great many rings (having rings of quotients of the given type), which are neither simple nor Noetherian. So what could one expect to discover about simple Noetherian rings? Here are some problems:

1:
Is every simple Noetherian ring isomorphic to a complete matrix ring over an integral doniain L? (= endomorphism ring of a finitely generated free L-modulel; free module=module with a basis).

2:
Is every simple Noetherian ring isomorphic to the endomorphism ring of a finitely generated projective module over a simple integral domain? (projective module = direct summand of a free module).

3:
Does every simple Noetherian ring with zero divisors have an idempotent (besides 0,1)?

As far as I know, these problems are still open; the answers are affirmative if the ring has minimum condition. In connection with problem 2, it is relatively easy to show that every simple Noetherian ring is isomorphic to the endomorphism ring of a finitely generated projective module over an integral domain of some sort.

There is a notion of "non-commutative Dedekind ring", and it is possible for a ring to be simldltaneously simple and Dedekind. For a simple Noetherian ring, the requirement is just that every left and right ideal should be a projective module; this condition is satisfied by all simple rings with minimum condition, and by some without it. For this class of rings the questions corresponding to 2 and 3 above have affirmative answers; the question corresponding to 1 is still open even in this case. The answer to 2 for this class gives a nice generalization of the Artin-Wedderburn' theorem. Here are the promised examples:

1:
Let F be a field of characteristic zero, L the F-algebra generated by x and y where xy - yx = 1. L is a simple Dedekind ring, and an integral domain. LÄFLÄF ... ÄFL is a simple Noetherian iritegral domain.

2:
Let K be a commutative Noetherian integral domain of characteristic zero, d: K --> K a derivation (i.e. a group homonlorphism with d(ab) = a.db + da.b) leaving no ideal invariant (except 0 and K). Let L be the ring generated by K and an indeterminate x satisfying xk - kx = dk for all k in K. Then L is a simple Noetherian integral domain. If K is a field, L is a principal ideal domain.

3:
Let L be any simple Noetherian integral domain, P a finitely generated projective L-module. The endornorphism ring of P is a simple Noetherian ring, which has zero-divisors in general. If P is free, we just get a complete matrix ring over L.