In [2] Brungs shows that every ring T between a principal (right and left) ideal domain R and its quotient field is a quotient ring of R. In this note we obtain similar results without assuming the ascending chain conditions. For a (right and left) Bezout domain R we show that T is a quotient ring of R which is again a Bezout domain; furthermore Tis a valuation domain if and only if T is a local ring.

The study of systems of conies and other algebraic curves was initiated in the middle of the nineteenth century by Cayley, Hesse, Cremona, and others. Most of the investigations from that time to the present have been concerned with extensions to algebraic varieties and systems of higher orders or dimensions, or with associated algebraic curves such as Jacobians and Hessians.

Recently Luedeman studied certain idempotent topologizing families of left ideals in semi-group rings AS which arise from such families of left ideals of A. Let ∑ be an idempotent topologizing family of left ideals in A and G a group, let ∑G denote the family of left ideals of AG which contain left ideals of the form LG, L ∈ ∑.

It is proved, for any uncountable cardinal λ, that a λ-complete Boolean ring is λ-self-injective. An example shows that the converse need not hold.

The problem of finding the largest set of nodes in a d-cube of side 3 such that no three nodes are collinear was proposed by Moser. Small values of d (viz., d ≤,3) resulted in elegant symmetric solutions. It is shown that this does not remain the case in 4 dimensions where at most 43 nodes can be chosen, and these must not include the center node.

Let R be an associative ring (not necessarily with identity).

R is a left П-ring if it has the following property: Let M be a finitely generated left R-module, N a submodule of M and ϕ:N→M an epimorphism. Then ϕ is an isomorphism.

A presentation of a group G is an exact sequence of groups

where F is a free group. Let l→S ⊆ F→G→1 be another presentation of G involving the same free group F.

Let A be an ideal of the commutative ring R with identity. There is a canonical homomorphism ϕ A from the polynomial ring R[X] onto (R/A)[X], obtained by reducing all coefficients modulo A. If f R[X], then we say that f is reducible (irreducible) modulo A if ϕ A (f) is reducible (irreducible) in (R/A)[X].

A line-coloring of a graph G is an assignment of colors to the lines of G so that adjacent lines are colored differently; an n-line coloring uses n colors. The line-chromatic number χ'(G) is the smallest n for which G admits an n-line coloring.

In 1959, Moser [4] posed the following problem: how should a pair of n-sided dice be loaded (identically) so that, on throwing the dice, the frequency of the most frequently occurring sum is as small as possible? This can be recast in the following form: determine for each n(≥1), the polynomial P n (x) which minimizes the maximum coefficient in the polynomial subject to the conditions that the coefficients of P n (x) are nonnegative and sum to unity.

On considère un opérateur U(x) qui transforme l'espace euclidien En en soimême, en général discontinu, et on étudie la convergence d'un processus itératif de la forme x p+1 = x p -μ U(x) (μ est une constante numérique positive). Processus de ce type, avec U(x) discontinu, se rencontrent par exemple à l'algorithme de relaxation pour la résolution des systémes d'inéquations [1], [2], de même qu'au calcul des polynômes de la meilleure approximation sur un ensemble fini de points [3].

In 1962, J. M. G. Fell [5] indicated the important role played by certain topological spaces which, though locally compact in a specialized sense, do not, in general, satisfy even the weakest separation axiom. He called them "locally compact". These were called "punktal kompakt" by Flachsmeyer [6] and to avoid confusion, we shall call them pointwise compact spaces.

Let 〈x〉 be an infinite cyclic group and R i 〈x〉 its group ring over a ring (with identity) R i , for i = l and 2. Let J(R i ) be the Jacobson radical of R i . In this note we study the question of whether or not R 1〈x〉≃R 2〈x〉 implies R 1≃R 2. We prove that this is so if Z i the centre of R i is semi-perfect and J(Z i 〈x〉) = J(Z i 〈)x〉 for i = l and 2. In particular, when Z i is perfect the second condition is satisfied and the isomorphism of group rings R i 〈x〉 implies the isomorphism of R i .

In a recent paper [3] Meir and Sharma introduced a generalization of the Sα- method of summability. The elements of their matrix, (a nk ), are defined by

(1)

where is a sequence of complex numbers. if 0 < αj < l for each j = 0, 1, 2,… then a nk ≥0 for each n = 0, 1, 2,… and k = 0,1,2,…

This paper is a continuation of the two series, Eigenvalues of Sums I, II, III [2], [3], [4] and Singular Values of Products I, II, III [5], [6], [7].

Let (Ω,ℱ, P) be a probability space. Let R denote the set of real numbers and the set of all random variables defined on Ω. Throughout this work, random variables which differ only on a set of probability zero will be considered identical. EX represents, as usual, the expectation of .

In this note we generalize the following classical theorem: If μ and ν are finite real-valued measures such that ν(A) = 0 implies μ(A) = 0, then for every ε > 0, there exists δ > 0 such that μ(A)<ε whenever ν(A)< δ.

An m-bounded extension of a topological space is an m-bounded space which contains the original as a dense subspace. m-bounded spaces have been studied by Gulden, Fleischman, and Weston [4], Saks and Stephenson [6], and Woods [8].

This note, motivated by [2], [3], and [4], is devoted to an investigation of properties related to equicontinuity in function spaces of topological spaces. In §2, we study the property (G) defined in [3], and the regularity defined in [4]. A sufficient condition for the simultaneous continuity of a function of two variables, which is analogous to a well known result in equicontinuity, is given at the end of the section. In §3, we relate the regularity with the semi-equicontinuity defined in [2], by localizing the semi-equicontinuity in an obvious way which leads us to weaken some of the hypotheses used in [2]. By the way of constructing an example, we also obtained a sufficient condition for a regular semitopological group to be a topological group.

In [1], Willard proves the following

If a regular paracompact space X has a dense Lindelöfsubspace, then X is Lindelöf.

Willard notes that the above is a generalization of the standard theorem: A separable paracompact space is Lindelöf. Actually, it is a standard fact ([2, p. 24]) that a separable metacompact space is Lindelöf. Moreover, one discovers that if a separable space X is such that each open cover of X has a point-countable open refinement, then X is Lindelöf.