This paper attempts to generalize a property of regular rings, namely,I2=I for every right (left) ideal. Rings with this property are called right (left) weakly regular. A ring which is both left and right weakly regular is called weakly regular. Kovacs in [6] proved that, for commutative rings, weak regularity and regularity are equivalent conditions and left open the question whether for arbitrary rings the two conditions are equivalent. We show in §1 that, in general weak regularity does not imply regularity. In fact, the class of weakly regular rings strictly contains the class of regular rings as well as the class of biregular rings.

In this paper the author characterizes images of class- spaces (as defined by Ishii, Tsuda and Kunugi, Proc. Japan Acad. 44, (1968), 897-903) under almost-open maps, bi-quotient maps, pseudoopen maps and quotient maps.

In the past three decades the problem of generating (balanced incomplete block) designs by difference sets has received much attention. Bose [2] gave the two "fundamental theorems of the method of differences". Bose, Sprott [9], Lehmer [7], Chowla [4], Takeuchi [10] and others have given specific classes of difference sets.

Let(X1, Y1), (X2, Y2),…, (Xn, Yn) be n mutually independent pairs of random variables with absolutely continuous (hereafter, a.c.) pdf given by

(1)

where f(ρ) denotes the conditional pdf of X given Y, g(y) the marginal pdf of Y, e(ρ)→ 1 and b(ρ)→0 as ρ→0 and,

(2)

We wish to test the hypothesis

(3)

against the alternative

(4)

For the two-sided alternative we take — ∞< b < ∞. A feature of the model (1) is that it covers both-sided alternatives which have not been considered in the literature so far.

H. T. Croft's Research Problems, 1967, contains the following problem due to J. H. Conway. "Let A = {a1, a2,…, aN} be a finite set of integers, and define

and

Prove that A—A always has more members than A+A, unless A is symmetric about 0."Marica in [1] showed that the conjecture is false for the set A = {1, 2, 3, 5, 8, 9, 13, 15, 16}. In this case A+A has 30 elements and A—A has 29 elements.

Aczel has conjectured, [1, p. 448], the possibility of developing a net theory for structures more general than quasigroups. Steps in this direction have been taken by Havel who considers nets associated with multigroupoids [2], The work presented here introduces a generalization of 3-nets and their algebraization which is wide enough to encompass most algebraic structures based on a single binary operation.

Let p be an odd prime, n an integer not divisible by p and α a positive integer. For any integer h with (h,pα)=l, is defined as any solution of the congruence (mod,pα). The Kloosterman sum Ap α(n) (see for example [4]) is defined by

(1.1)

where the dash (') indicates that the letter of summation runs only through a reduced residue system with respect to the modulus.

In answer to a question raised by Leo Moser, A. Meir proved some years ago that every plane arc of unit length lies in some closed semidisk of radius ½. His elegant, unpublished argument is reproduced here with his kind permission.

The Hermite conjugate Poisson integral of a given f ∊ L1(μ), dμ(y)= exp(—y2) dy, was defined by Muckenhoupt [5, p. 247] as

where

If the Hermite conjugate function operator T is defined by (Tf) a.e., then one of the main results of [5] is that T is of weak-type (1, 1) and strongtype (p,p) for all p>l.

If f is a suitable meromorphic function, then by a classical technique in the calculus of residues, one can evaluate in closed form series of the form,

Suppose that a(n) is an arithmetical function. It is natural to ask whether or not one can evaluate by contour integration

(1.1)

where f belongs to a suitable class of meromorphic functions. We shall give here only a partial answer for a very limited class of arithmetical functions.

Let a particle move along a straight line a unit distance during every interval of time τ. During the first interval τ it moves to the right with probability ρ1 and to the left with probability ρ2 = 1 - ρ1. Thereafter at the end of each interval τ, the particle with probability p continues its motion in the same direction as in the previous step and with probability q = l - p reverses it.

A general categorical construction is described which has as special cases the construction of the Brauer group of a field and the construction of the Witt ring of a field of characteristic ≠2.

Suppose X and F are nxn matrix solutions of the n X n matrix differential equation

(1)

such that

(2)

where J is some interval.

An inverse limit of finite groups has been called in the literature a pro-finite group and we have extensive studies of profinite groups from the cohomological point of view by J. P. Serre. The general theory of non-abelian modules has not yet been developed and therefore we consider a generalization of profinite abelian groups. We study inverse systems of discrete finite length R-modules. Profinite modules are inverse limits of discrete finite length R-modules with the inverse limit topology.

Let L/k be a Galois extension of fields with group G and let A be the category of k-algebras isomorphic to finite products of finite field subextensions of L/k. It is known that, with appropriately defined covers, A is dual to the underlying category of a Grothendieck topology T [5, Ch. I, Theorem 4.2] and that (strict) cohomological dimension of G may be characterized via TCech cohomology with coefficients in either additive (product-preserving) functors or sheaves [5, Ch. I, Theorems 4.3 and 5.9].

The purpose of this note is to present a characterization of m-paracompact normal spaces in terms of ω-mappings into m-separable metric spaces; this result is almost contained in Morita's original paper [2] and is implicitly contained in Shapiro's thesis paper [5].

Satyanarayana establishes in [6] that a semiprime right selfinjective ring with ACC on annihilator right ideals is semisimple Artinian, thereby extending a similar result of Koh [5] for prime rings. A theorem of Faith [3, Theorem 5.2], shows that the annihilator chain condition on either side implies that a right selfinjective semiprime ring is semisimple Artinian. Noting that any selfinjective ring has torsionless injective envelope we consider the possibility of replacing selfinjectivity by torsionless together with an annihilator condition.

Given a pair of functors F, G:A→B, Lambek defines [1] the subequalizing category, E of (F, G) as the category with objects, ordered pairs (A, b) with A ∈ |A| and b:FA→GA a morphism of B. The morphisms of E from (A, b) to (A′, b′) are ordered triples (b, a, b′) where a:A→A′ is a morphism of A and G(a)b = b′F(a).

It is well known (see [3](1)) that no continuum (i.e. compact, connected, Hausdorff space) can be written as a countable disjoint union of its (nonvoid) closed subsets. This result can be generalized in two ways into the setting of locally compact, connected, Hausdorff spaces. Using the one point compactification of a locally compact, connected, Hausdorff space X one can easily show that X cannot be written as a countable disjoint union of compact subsets. If one makes the further assumption that X is locally connected, then one can show that X cannot be written as a countable disjoint union of closed subsets.(2)