Throughout this note, rings are associative with identity element but are not necessarily commutative. Let R be a left and right Noetherian ring which has an Artinian (classical) quotient ring. It was shown by S. M. Ginn and P. B. Moss [2, Theorem 10] that there is a central idempotent element e of R such that eR is the largest Artinian ideal of R. We shall extend this result, using a different method of proof, to show that the idempotent e is also related to the socle of R/N (where N, throughout, denotes the largest nilpotent ideal of R) and to the intersection of all the principal right (or left) ideals of R generated by regular elements (i.e. by elements which are not zero-divisors). There are many examples of left and right Noetherian rings with Artinian quotient rings, e.g. commutative Noetherian rings in which all the associated primes of zero are minimal together with full or triangular matrix rings over such rings. It was shown by L. W. Small that if R is any left and right Noetherian ring then R has an Artinian quotient ring if and only if the regular elements of R are precisely the elements c of R such that c + N is a regular element of R/N (for further details and examples see [5] and [6]). By the largest Artinian ideal of R we mean the sum of all the Artinian right ideals of R, and it was shown by T. H. Lenagan in [3] that this coincides in any left and right Noetherian ring R with the sum of all the Artinian left ideals of R.

The present paper deals with relations between flat overrings and quotient rings. Weare mainly concerned with Richman's results [10] on flat overrings and withthose of Davis [2], Gilmer [3], Gilmer and Heinzer [4], Gilmer and Ohm [5], and Mott [8], on rings with the QR propertyand with the property (#) defined in Section 1. Some of their results are generalized, and it is shown that certain theorems, which at first glance seem to have nothing in common, are in fact particular cases of a single more general theorem.

In the literature there are several generalisations to non-commutative rings of the notion of a unique factorisation domain from commutative algebra. This paper follows in the spirit of [1, 3] and is set in the context of Noetherian rings. In [3], A. W. Chatters and the author denned a Noetherian UFR (unique factorisation ring) to be a prime Noetherian ring R in which every non-zero prime ideal contains a prime ideal generated by a non-zero normal element p, that is, by an element p such that pR = Rp. The class of Noetherian UFRs includes the Noetherian UFDs studied by Chatters in [1], while a commutative Noetherian ring is a UFR if and only if it is a UFD in the usual sense. For a Noetherian UFR R, the following are simple consequences of the definition:

(i) every non-zero ideal of R contains a non-zero normal element;

(ii) the set N(R) of non-zero normal elements of R is a unique factorisation monoid in the sense of [4, Chapter 3].

The aim of this note is to generalize to an arbitrary partially ordered set (poset) (P, ≦) the standard lattice results on the Jordan–Dedekind Chain Condition (abbreviated hereafter to J.D.C.C.). Birkhoff [1] defines semimodularity for a lattice L by

(ξ) if x, y cover a and x # y, then x ∨ y covers x and y.

Let T be a linear operator acting in a Banach space X. It has been shown by Smart [5] and Ringrose [3] that, if X is reflexive, then T is well-bounded if and only if it may be expressed in the form

where {E(λ)} is a suitable family of projections in X and the integral exists as the strong limit of Riemann sums.

Motivated by a problem in mathematical economics [4] Gretsky and Ostroy have shown [5] that every positive operator T:L1[0, 1] → c0 is a Dunford-Pettis operator (i.e. T maps weakly convergent sequences to norm convergent ones), and hence that the same is true for every regular operator from L1[0, 1] to c0. In a recent paper [6] we showed the converse also holds, thereby characterizing the D–P operators by this condition. In each case the proof depends (as do so many concerning D–P operators on Ll[0, 1]) on the following well-known result (see, e.g., [2]): If μ is a finite measure, an operator T:L1(μ) → E is a D–P operator is compact, where i:L∞(μ) → L1(μ) is the canonical injection of L∞(μ) into L1(μ). If μ is not a finite measure this characterization of D–P operators is no longer available, and hence results based on its use (e.g. [5], [6]) do not always have straightforward extensions to the case of operators on more general L1(μ) spaces.

To what extent is the structure of a Lie-algebra L over a field F determined by the bilinear form

on L that is derived from a matrix representation

of L with finite degree d(Δ) by forming the trace of the matrix products

Such a bilinear form is a function with two arguments in L, values in F and the properties:

According to Bourbaki [1, pp. 62–63, Exercise 11], a left (resp. right) A-module M is said to be pseudo-coherent if every finitely generated submodule of M is finitely presented, and is said to be coherent if it is both pseudo-coherent and finitely generated. This Bourbaki reference contains various results on pseudo-coherent and coherent modules. Then, in [1, p. 63, Exercise 12], a ring which as a left (resp. right) module over itself is coherent is said to be a left (resp. right) coherent ring, and various results on and examples of coherent rings are presented. The result stated in [1, p. 63, Exercise 12a] is a basic theorem of [2] and first appeared there. A variety of results on and examples of coherent rings and modules are presented in [3].

The Laplace transform of a function f(t) ∈ L(0, ∞) is defined by the equation

and its Hankel transform of order v is defined by the equation

The object of this note is to obtain a relation between the Laplace transform of tμf(t) and the Hankel transform of f(t), when ℛ(μ) > − 1. The result is stated in the form of a theorem which is then illustrated by an example.

Interest in the ranges of Lyapunov transformations began with Taussky in [5]. Recently in a series of papers, Loewy has studied the ranges of Lyapunov transformations on matrices. In particular in [2] and [3], the following result was obtained.

Definitions and introduction. Let Ц = {Ui|i ∈ I} be a system of subsets of a normal topological space R; i.e. a mapping from the index set I into the set of all subsets of R. The order of a point x is the number of distinct member sets of Ц which contain x, and is denoted by x: Ц; the sets Ui are here considered distinct if they have distinct indices. Thus x: Ц is the number of indices i for which x ∈ Ui; ν(Ц) = max {x: Ц | x ∈ R} is called the order of the system Ц. If every point has an (open) neighbourhood meeting only finitely many members of Ц, then Ц is said to be locally finite.

When Ramanujan died in 1920 he left behind three notebooks containing statements of a few thousand theorems, mostly without proofs. The second notebook is an enlarged edition of the first, and the third is short and fragmentary. Thus our primary attention may be directed toward the second notebook. In the decade following Ramanujan's death, G. N. Watson and B. M. Wilson agreed to perform the enormous task of editing the notebooks. Unfortunately, this task was never completed, possibly, in part, due to the premature death of Wilson in 1935. In 1957, a photostat edition [19] of the notebooks was published, but no editing whatsoever was undertaken.

Introductory Remarks. Recently a number of studies (Chen & Saffman [2], Jones & Toland [7,11], Hogan [5]) have been made of periodic capillary-gravity waves which form the free surface of an ideal fluid contained in a channel of infinite depth. However, little work appears to have been done on the corresponding problem when the depth is finite. The most significant contributions appear to be those of Reeder & Shinbrot [9], Barakat & Houston [1] and Nayfeh [8] all of whom confined themselves to Wilton ripples (see §1.3). Yet there are sound reasons why such a study should be made. For quite apart from the unsolved problem regarding the type of capillary-gravity waves which may occur at finite depths, the consideration of the finite depth problem may be regarded as a first step in the study of solitary capillary-gravity waves. In this paper, a new integral equation for the infinite depth problem, due to J. F. Toland and the author, is adapted to be of use in tackling the finite depth problem. Using this we obtain results for the exact equations of motion which answer rigorously the questions of existence and multiplicity of small amplitude solutions of the periodic capillary-gravity wave problem of finite depth.

Recently some inversion integrals for integral equations involving Legendre, Chebyshev, Gegenbauer and Laguerre polynomials in the kernel have been obtained [1, 2, 3, 5, 6]. In this note, two inversion integrals for integral equations involving Whittaker's function in the kernel are obtained. We shall make use of the following known integral [4, p. 402]

The results of this note are based on the following two integrals, which are derived from (1) by writing u – t = (v – t)x.

for m + 1 > 2v > – 1;

for m + 1 > 2v > – 1.

The formulae to be proved are

where m is a positive integer, p ≧ q + 1, R(mar + k) > 0, r = 1, 2, …, p, and | amp z |ππ. For other values of p and q the result holds if the integral is convergent.

Let ℤ and ℤ[i] have their usual meaning. Let Yo denote the noncommutative ring of integral quaternions, that is the set of all elements a + bi + cj + dk with a, b, c, d ∈ ℤ and where i, j and k together with the number 1 are the four units of the system of quaternions.

The Laplace transform

has been generalised by Varma [4] by the relation

which reduces to (1.1) when k = -m + ½ by virtue of the identity

We shall define πk, m, λ (p) by the relation

The object of this paper is to obtain some recurrence formulae and series for πk, m, λ (p) and to use them to obtain recurrence formulae and series for MacRobert's E-function.

Recently, Saleh [3] claimed to have solved ‘a long standing open question’ in topology; namely, he proved that every almost continuous function is clousure continuous (= θ = continuous). Unforunately, this problem was settled long time ago and even a better result is known. Consider the following implications: Cont. ⇒ Almost cont. ⇒ Almost α-cont.⇒ η-cont. ⇒.θ-cont. ⇒ Weakley cont.