Throughout this paper all rings are assumed commutative with identity. Among integral domains, Dedekind domains are characterized by the property that every ideal is a product of prime ideals. For a history and proof of this result the reader is referred to Cohen [2, pp. 31–32]. More generally, Mori [5] has shown that a ring has the property that every ideal is a product of prime ideals if and only if it is a finite direct product of Dedekind domains and special principal ideal rings (SPIRS). Rings with this property are called general Z.P.I.-rings.

In this note I show that

where Jdenotes the Bessel function of the first kind of the orders and arguments indicated, n = 0, 1, 2, 3, … and the real parts of both μand v exceed — 1. This is a generalization of Sonine's first finite integral [1, p. 373] to which it reduces in the special case n = 0.

In § 2 a number of infinite series of E-functions are summed by expressing the E-functions as Barnes integrals and interchanging the order of summation and integration.

The Barnes integral employed is

where | amp z | < π and the integral is taken up the η;-axis, with loops, if necessary, to ensure that the origin lies to the left of the contour and the points α1, α2,… αp to the right of the contour. Zero and negative integral values of the α's and p's are excluded, and the α's must not differ by integral values. When p < q + 1 the contour is bent to the left at each end.

Our main purpose is to describe those separated locally convex spaces which can serve as domain spaces for a closed graph theorem in which the range space is an arbitrary Banach space of (linear) dimension at most c, the cardinal number of the real line R. These are the δ-barrelled spaces which are considered in §4. Many of the standard elementary Banach spaces, including in particular all separable ones, have dimension at most c. Also it is known that an infinite dimensional Banach space has dimension at least c (see e.g. [8]). Thus if we classify Banach spaces by dimension we are dealing, in a natural sense, with the first class which contains infinite dimensional spaces.

If R is a local (Noetherian) ring, it is well known that R is regular if and only if its completion is regular. It is the purpose of this note to show that a similar result is true for Noether lattices.

In this note we formally solve the following dual integral equations:

where h is a constant and the Fourier cosine transform of u–1 φ(u) is assumed to exist. These dual equations arise in a crack problem in elasticity theory.

It is sometimes possible to reconstruct semigroups from some of their homomorphic images. Some recent examples have been the construction of bisimple inverse semigroups from fundamental bisimple inverse semigroups [9], and the construction of generalized inverse semigroups from inverse semigroups [12].

One of the fundamental results of representation theory is the identification of the irreducible representations of a semisimple group by their dominant weights [3]. The purpose of this paper is to establish similar results for a class of reductive algebraic monoids.

Let k be an algebraically closed field. An algebraic monoid is an affine algebraic variety M defined over k, together with an associative morphism m:M × M → M and a two-sided unit 1 ∈ M for m.

Linear spaces on which both an order and a topology are defined and related in various ways have been studied for some time now. Given an order on a linear space it is sometimes possible to define a useful topology using the order and linear structure. In this note we focus on a special type of space called a linear lattice and determine those lattice properties which are both necessary and sufficient for the existence of a classical norm, called an M-norm, for the lattice. This result is a small step in a program to determine which intrinsic order properties of an ordered linear space are necessary and sufficient for the existence of various given types of topologies for the space. This study parallels, in a certain sense, the study of purely topological spaces to determine intrinsic properties of a topology which make it metrizable and the study of the relation between order and topology on spaces which have no algebraic structure, or. algebraic structures other than a linear one.

In the study of certain prime Noetherian rings it is natural to consider the set C of elements which are regular modulo all height-1 prime ideals of R. For R commutative, this set C is simply the set of units. In general this is not the case, though with certain additional conditions we can state non-commutative versions of the Principal Ideal Theorem.

An interesting concept of semigroups (and also rings) of (left) quotients, based on the notion of group inverse in a semigroup, was developed by J. B. Fountain, V. Gould and M. Petrich, in a series of papers (see [5]-[12]). Among the most interesting are semigroups having a semigroup of (left) quotients that is a union of groups. Such semigroups have been widely studied. Recall from [3] that a semigroup has a group of left quotients if and only if it is right reversible and cancellative. A more general result was obtained by V. Gould [10]. She proved that a semigroup has a semilattice of groups as its semigroup of left quotients if and only if it is a semilattice of right reversible, cancellative semigroups. This result has been since generalized by A. El-Qallali [4]. He proved that a semigroup has a left regular band of groups as its semigroup of left quotients if and only if it is a left regular band of right reversible, cancellative semigroups. Moreover, he proved that such semigroups can be also characterised as punched spined products of a left regular band and a semilattice of right reversible, cancellative semigroups. If we consider the proofs of their theorems, we will observe that the principal problem treated there can be formulated in the following way: Given a semigroup S that is a band B of right reversible, cancellative semigroups Si, i ε B, to each Si, we can associate its group of left quotients Gi. When is it possible to define a multiplication of such that Q becomes a semigroup having S as its left order, and especially, that Q becomes a band B of groups Gi, i E B?Applying the methods developed in [1] (see also [2]), in the present paper we show how this problem can be solved for Qto become a strong band of groups (that is in fact a band of groups whose idempotents form a subsemigroup, by [16, Theorem 2]. Moreover, we show how Gould's and El-Quallali's constructions of semigroups of left quotients of a semilattice and a left regular band of right reversible, cancellative semigroups, can be simplified.

Recently, P. Kaplan and K. S. Williams [10] considered (as an example) the representation of primes by binary quadratic forms of discriminant –768. These forms fall into 4 genera, each consisting of two classes. In particular, they considered the forms

F=3X2+642 and G = 12X2+12XY+19Y2.

It follows from genus theory (as explained in [10]) that every prime p ≡ 19 mod 24 is represented by exactly one of the forms F and G. Based on numerical data, they conjectured that a prime p ≡ 19 mod 24 is represented by

where

Vo = 2, V1 = -4, Vn+2=-4Vn+1 -Vn (n∨0).

Isotropes play a distinguished rôle in the algebra of spinors. Let V be an even-dimensional real vector space equipped with an inner product Bof arbitrary signature. An isotrope of (V, B) is a subspace of the complexification Vc on which Bc is identically zero. Denote by ρ the spin representation of the complex Clifford algebra C(Vc, Bc) on a space S of spinors.

The asymptotic behavior of the solutions of ordinary nonlinear differential equations will be considered here. The growth of the solutions of a differential equation will be discussed by establishing criteria to determine when the differential equation does not possess a solution that is an element of the space Lp(0, ∞)(p ≧ 1).

Let(V, ≧, ‖ · ‖) be a Banach lattice, and denote V\{0} by V'. For the definition of a Banach lattice and other undefined terms used below, see Vulikh [4]. Leader [3] shows that, if norm convergence is equivalent to order convergence for sequences in V, then the norm is equivalent to an M-norm. By assuming the equivalence for nets in V we can strengthen this result.

In the course of some recent work on Fourier series [5, 6] I had occasion to use a number of integral inequalities which were generalizations or limiting cases of known results. These inequalities may perhaps have other applications, and it seems worth while to collect them together in a separate note with one or two further results of a similar nature.

For any number k, used as an index (exponent), and such that K > 1, we write k' = k′(k–1), so that k and k′ are conjugate indices in the sense of Hölder's inequality.