This paper introduces a class of domains which we hope to show merits some attention.

Definition. The domain R is said to be a distinguished domain if for any 0 ≠ z ∈ K, the quotient field of R, (1 : z) does not consist entirely of zero divisors modulo (1 : z–l). (Note: Here we use the fact that a zero module has no zero divisors. Thus if z–l ∈ R, so that (1 : z–l) = R, then the condition holds trivially.)

Section 1 of this paper gives numerous examples of distinguished domains, foremost among them being Krull domains and Prufer domains. In fact Prüfer domains are shown to be exactly those distinguished domains whose prime lattice forms a tree. Other distinguished domains can be constructed by the D + M construction. It is shown that distinguished domains are integrally closed but the converse fails.

Let R be a commutative ring, C a finite abelian group, S a Galois extension of R with group C, in the sense of [1]. Viewing S as an RC-module defines the Picard invariant map [4] from the Harrison group Gal (R, C) of isomorphism classes of Galois extensions of R with group C to CI (RC), the class group of RC. The image of the Picard invariant map is known to be contained in the subgroup h Cl (RC) of primitive elements of CI (RC) (for definition see below). Characterizing the image of the Picard invariant map has been of some interest, for the image describes the extent of failure of Galois extensions to have normal bases.

Let R be the ring of integers of an algebraic number field K.

Let Σak be an infinite series of real numbers and let π be a permutation of N, the set of positive integers. The series Σaπ(k) is then called a rearrangement of Σak. A classical theorem of Riemann states that if Σak is a conditionally convergent series and s is any fixed real number (or ± ∞), then there is a permuation π such that Σaπ(k) = s. The problem of determining those permutations that convert any conditionally convergent series into a convergent rearrangement (such permuations are called convergence preserving) has received wide attention (see, for example [6]). Of special interest is a paper by P. A. B. Pleasants [5] in which is shown that the set of convergence preserving permutations do not form a group.

In this paper we consider questions similar to those above, but for rearrangements of divergent series of positive terms. We establish some notation before stating the precise problem.

Let KG be the group ring of a group G over a field K, and U its group of units. Given a group H, we shall denote by ξ(H) the center of H and by T(H) the set of all its torsion elements.

The following question appears in [5, p. 231]: When is Un ⊂ ξ (U), for some n? It was considered by G. Cliff and S. K. Sehgal in [1], where G is assumed to be a solvable group. A complete answer at characteristic zero is given there. Also they obtain partial results at characteristic p ≠ 0, with certain restrictions on the exponent n.

Let L be a semilattice and let a ∊ L. We refer the reader to Definitions 2.2, 2.4, 2.5 and 2.12 below for the terminology. If L is a-implicative, let Ca be the set of a-closed elements of L, and let Da be the filter of a-dense elements of L. Then Ca is a Boolean algebra. If a = 0, then C0 and D0 are the usual closed algebra and dense filter of L. If L is a-admissible and f : Ca × Da → Da is the corresponding admissible map, we can form a quotient semilattice Ca × D0f. In case a = 0, Murty and Rao [4] have shown that C0 × D0/f is isomorphic to L, and hence that C0 × D0 is 0-admissible. In case L is in fact implicative, Nemitz [5] has shown that C0 × D0/f is isomorphic to L, and that C0 × D0/f is also implicative.

This note is concerned with a simple but rather surprising identity which emerged unexpectedly from the work of one of the authors on the characterisation of characters. Consider, for example, the seven partitions of 5. These are

(1) 5, 4 1, 3 2, 3 12, 22 1, 2 13, 15

and with each of these we can associate a product of factorials of the numbers of repetitions, respectively

(2) 1!, (1!)(1!), (1!)(1!), (1!)(2!), (2!)(1!), (1!)(3!), 5!

It is then seen that the product of all the numbers occurring in (1) coincides with that of all the numbers in (2).

Generally, for any particular natural number n, the partitions can be written in the form

1a1 2a2 3a3…,

in which ak is the frequency of repetition of the part k, and are enumerated by the distinct sets {α} = {a1, a2, …,} with ak ≧ 0 and Σkak = n.

In this paper, we study the intersection pattern of families of convex sets. Since we only consider finite families, we may assume that the sets are also compact.

As an example, we consider families of 5 convex sets in R2 such that every two intersect and no three intersect. One such family that comes immediately to mind is that of 5 lines in general position. However, this is not the only family which exhibits this intersection pattern. Fig. 1 shows a family of 3 lines (sides of the large triangle) and 2 triangles (inscribed in the large triangle) that has this property.

We wish to consider the class of (0,1)-matrices with prescribed row and column sums. Let R = (r1, r2, …, rm) and S= (s1, s2, …, sn) be vectors with nonnegative integral entries and r1 + r2 + … + rm = s1 + s2 + … + sn. We define the class to be the set of m × n (0, 1)-matrices with ith row sum ri and jth column sum sj for 1 ≦ i ≦ m and 1 ≦ j ≦ n.

Gale and Ryser independently found simple necessary and sufficient conditions for to be nonempty [9 , 14]. From R, we form an m × n (0, 1)-matrix Ā as follows. The ith row sum of Ā is ri and the 1‘s are as far to the left as possible. Let be the jth column sum of A. We define the sequence to be the conjugate of the sequence (ri).

A theorem of Eisenbud, Griffith, and Robson states that if R is hereditary and noetherian (on both the left and right) then every proper homomorphic image of R is a generalized unserial ring (see, for example, [3, p. 244]). Singh [11, p. 883] states a converse: If R is a right bounded, noetherian prime ring, all of whose proper homomorphic images are generalized uniserial rings, then (every divisible right R-module is injective, so) R is right hereditary. (Actually, Singh omitted the clearly necessary “bounded” condition.) Singh's theorem generalizes results of [9, Proposition 15], [2, Theorem 2.1], and [8], about commutative rings.

We will call a semi-prime ring R essentially right bounded if each essential right ideal contains a two-sided ideal which is essential as a right ideal. In case R is prime, “essentially right bounded” coincides with “right bounded”.

1. Introduction. Many problems involving prime numbers depend on estimating sums of the form ΣΛ(n)f(n), for appropriate functions f(n), (here, as usual, Λ(n) is the von Mangoldt function). Three distinct general methods have been used to estimate such sums. The earliest is due to Vinogradov (see [13, Chapter 9]); the second involves zerodensity bounds for Dirichlet L–functions (see [8, Chapters 15 and 16] for example); and the third, due to Vaughan (see [12] for example) uses an arithmetical identity as will be explained later. The second and third methods are much simpler to apply than the first. On the other hand Vinogradov's technique is at least as powerful as Vaughan's and occasionally more so. In many cases Vaughan's identity yields better bounds than the use of zero–density estimates, but sometimes they are worse. The object of this paper is to present a simple extension of Vaughan's method which is essentially as powerful as any of the techniques mentioned above, to discuss its general implications, and to apply it to the proof of the following result of Huxley [4], which has previously only been within the scope of the zero density method.

0. Introduction. A point p ∈ βX\X is called a remote point of X if P ∉ clβXA for each nowhere dense subset A of X. If X is a topological sum Σ{Xn : n ∈ ω} we call nice if {n : F ∩ Xn = ∅} is finite for each . We call remote if for each nowhere dense subset A of X there is an with F ∩ A = ∅ and n-linked if each intersection of at most n elements of is non-empty.

Let H be a graded commutative algebra with a nice set of algebra generators. Let H also be a comodule over a Hopf algebra A. In Section 2 we give conditions under which certain of these generators of H can be rechosen to be primitive. In addition we give explicit formulas expressing these primitive generators in terms of the original set of generators.

In Section 3 we apply the theory of Section 2 to the mod p homology of the Thorn spectra MO, MU and MSp. In particular we give two explicit descriptions of the image of the Hurewicz homomorphism for MO. One of these makes explicit the recursive computation of E. Brown and F. Peterson [1].

In Section 4 we give a variation of the theory of Section 2 which computes primitive generators of certain Hopf algebras. This theory is applied to study the primitive elements of H*(BU) and H*(SO; Z2).

Let R = ⊕α∊гRα be an integral domain graded by an arbitrary torsionless grading monoid Γ. In this paper we consider to what extent conditions on the homogeneous elements or ideals of R carry over to all elements or ideals of R. For example, in Section 3 we show that if each pair of nonzero homogeneous elements of R has a GCD, then R is a GCD-domain. This paper originated with the question of when a graded UFD (every homogeneous element is a product of principal primes) is a UFD. If R is Z+ or Z-graded, it is known that a graded UFD is actually a UFD, while in general this is not the case. In Section 3 we consider graded GCD-domains, in Section 4 graded UFD's, in Section 5 graded Krull domains, and in Section 6 graded π-domains.

In our paper [3] we considered four nniteness conditions for an orthomodular lattice (abbreviated: OML) L and conjectured their equivalence. The only question left open in that paper was whether an OML L which can be covered by finitely many blocks (maximal Boolean subalgebras) has only finitely many blocks. In this paper we give an affirmative answer to this question, in fact, we prove the slightly stronger result:

Let R > 1 and denote by ℰR the ellipse

(1)

If Pn is a polynomial of degree at most n such that

(2)

then ([2]; also see [13, p. 337] and [9, p. 158, Prob. No. 270])

(3)

The standard proof of this well known result runs as follows. The function

(4)

is entire and in view of (2) we have

Hence by the maximum modulus principle

(5)

which is clearly equivalent to (3).

Following the notation and the definitions in [1], let L(K) be the Chevalley group of type L over a field K, W the Weyl group of L and h the Coxeter number, i.e., the order of Coxeter elements of W. In a letter to the author, John McKay asked the following question: If h + 1 is a prime, is there an element of order h + 1 in L(C)? In this note we give an affirmative answer to this question by constructing an element of order h + 1 (prime or otherwise) in the subgroup Lz = 〈xτ(1)|r ∈ Φ〉 of L(K), for any K.

Our problem has an immediate solution when L = An. In this case h = n + 1 and the (n + l) × (n + l) matrix

has order 2(h + 1) in SLn+1(K). This seemingly trivial solution turns out to be a prototype of general solutions in the following sense.

Let dndn–1 … d0 be the b-ary representation of a positive integer N. Call the polynomial obtained from N base b. In the case the base is 10, f(x) will be called the polynomial obtained from N. Pólya and Szegö attribute the following theorem to A. Cohn [2, b. 2, VIII, 128]:

THEOREM 1. A polynomial obtained from a prime is irreducible.

This theorem was generalized in two different ways by John Brillhart, Andrew Odlyzko, and myself [1]. One way was by proving the theorem remains true regardless of the base being used. The second way was by permitting the coefficients of f(x) to be different from digits. Thus, for example, if , where 0 ≦ dk ≦ 167 for all k, and if f(10) is prime, then f(x) is irreducible. In this paper, Theorem 1 will be generalized in another way by considering composite N.

There are two principal kinds of input data for infinite loop space theory, namely E∞ spaces à la Boardman-Vogt [3] and May [7] and Γ-spaces à la Segal [14]. May and Thomason [13] introduced a common generalization and used it to prove the equivalence of the output obtained from these two kinds of input.

This suggests that any invariants of one kind of input should have analogs for the other. Homology operations are among the most basic invariants of E∞ spaces, and we here establish the analogous invariants for Γ-spaces. The definition is transparently obvious from the point of view of the common generalization but is at first sight rather surprising and unnatural from the point of view of Γ-spaces alone. Probably for this reason, there is no hint of the possibility of a direct definition of homology operations for Γ-spaces in the literature.

Le but de cet article est de montrer qu'il n'y a pas de relation étroite entre ces trois concepts.

Soit K une partie compacte du plan complexe fini, et soit A(K) (resp. a(K)) l'ensemble des fonctions continues sur K et holomorphes (resp. harmoniques) à l'intérieur K0 de K. Nous dirons que K est un ensemble d'approximation holomorphe (resp. harmonique) si toute fonction de A(K) (resp. a(K)) peut être approchée uniformément sur K par des fonctions holomorphes (resp. harmoniques) sur K, i.e., dans un voisinage de K.

Pour un voisinage fixé Ω de K, le théorème de Mergelian nous donne des conditions nécessaires et suffisantes, purement topologiques, pour que toute fonction de A(K) puisse être approchée uniformément sur K par des fonctions holomorphes dans Ω. D'autre part, on sait (par exemple [4]) qu'il n'est pas possible de caractériser topologiquement les ensembles d'approximation holomorphe.

Let G be a bounded, open, connected, non-empty subset of the complex plane C. We put the usual two dimensional (Lebesgue) area measure on G and consider the Hilbert space L2(G) that consists of the complex-valued, measurable functions defined on G that are square integrable. The inner product on L2(G) is given by the norm ‖h‖2 of a function h in L2(G) is given by ‖h‖2 = (∫G|h|2)1/2.

The Bergman space of G, denoted La2(G), is the set of functions in L2(G) that are analytic on G. The Bergman space La2(G) is actually a closed subspace of L2(G) (see [12 , Section 1.4]) and thus it is a Hilbert space.

Let G denote the closure of G and let C(G) denote the set of continuous, complex-valued functions defined on G.