Let R be a ring with 1 and let Q denote the maximal left quotient ring of R [6]. In a recent paper [12], Wei called a (left). R-module M divisible in case HomR (Q, N)≠0 for each nonzero factor module N of M. Modifying the terminology slightly we call such an R-module a Q-divisible R-module. As shown in [12], the class D of all Q-divisible modules is closed under factor modules, extensions, and direct sums and thus is a torsion class in the sense of Dickson [5].

Twenty-five years ago McCoy published a characterization of commutative subdirectly irreducible rings. This result was generalized by Thierrin to duo rings with the word “field” which appeared in McCoy's theorem replaced by “division ring”. The purpose of this note is to give another generalization in which the words “division ring” will be replaced by “simple ring with 1 ”. The techniques resemble those of McCoy and Thierrin.

In [2] we showed how, for a certain class of outer measures on a metric space, a measurable cover could be constructed for each subset A of the space. The function is a closure operator, and in this note some of the properties of the resulting topology are investigated. In particular, we obtain a sufficient condition for the space to be connected.

The following theorem is well known [1, p. 66]: Suppose that, in a ballot, candidate P scores p votes and candidate Q scores q votes, where p > q. The probability that throughout the counting there are always more votes for P than for Q, equals (p-q)/(p+q).

In [3, p. 77] Ryser notes the importance of the minimum of the permanent function on the class of (0, 1)-matrices having exactly k ones in each row and column. In [4] a lower bound was found for the minimum of the permanent on the class Λn of n × n (0, 1)-matrices with exactly three l's in each row and column. The purpose of our work is to improve this result, in particular we show that minA ∊ Λn (per A) ≥ 3(n-1).

After the publication of the important paper by Rota [9] on Möbius functions a large number of papers have appeared in which the ideas are applied or generalized in various directions, the papers by Crapo [3], Smith [10] and Tainiter [11] are some of them. The theory of Möbius functions is now recognized as a valuable tool in combinatorial and arithmetical research.

Sanderson (Canad. Math. Bull., 8 (1965), 505–513), considering a nonempty collection Σ of left ideals of a ring R, with unity, defined the concepts of “Σ-injective module” and “Σ-essential extension” for unital left modules. Letting Σ be an idempotent topologizing set (called a σ-set below) Σanderson proved the existence of a “Σ-injective hull” for any unital left module and constructed an Utumi Σ-quotient ring of R as the bicommutant of the Σ-injective hull of RR. In this paper, we extend the concepts of “Σinjective module”, “Σ-essentialextension”, and “Σ-injective hull” to modules over arbitrary rings. An overring Σ of a ring R is a Johnson (Utumi) left Σ-quotient ring of R if RR is Σ-essential (Σ-dense) in RS. The maximal Johnson and Utumi Σ-quotient rings of R are constructed similar to the original method of Johnson, and conditions are given to insure their equality. The maximal Utumi Σquotient ring U of R is shown to be the bicommutant of the Σ-injective hull of RR when R has unity. We also obtain a σ-set UΣ of left ideals of U, generated by Σ, and prove that Uis its own maximal Utumi UΣ-quotient ring. A Σ-singular left ideal ZΣ(R) of R is defined and U is shown to be UΣ-injective when Z Σ(R) = 0. The maximal Utumi Σ-quotient rings of matrix rings and direct products of rings are discussed, and the quotient rings of this paper are compared with these of Gabriel (Bull. Soc. Math. France, 90 (1962), 323–448) and Mewborn (Duke Math. J. 35 (1968), 575–580). Our results reduce to those of Johnson and Utumi when 1 ∊ R and Σ is taken to be the set of all left ideals of R.

In contrast to the very complicated structure of the lattice of equational classes of commutative semigroups (see [5]), the lattice of equational classes of commutative monoids (semigroups with unit) is isomorphic with N × N* with a unit adjoined, where N is the lattice of natural numbers with the usual order and N* is the lattice of natural numbers ordered by division. (See [4].) However, the lattice of equational classes of commutative semigroups-with-zero is not so simple to describe.

In 1927 van der Waerden [6] proved that given positive integers k and l, there exists an integer W such that if 1, 2, …, W are partitioned into k or fewer classes, then at least one class contains an l-term arithmetic progression (l-progression). Let W(k, l), be the smallest such integer W. It would be of interest to find a reasonable upper estimate for W(k, l), say one that could be written down.

Recently, Ronveaux [11] has shown how to use a combination of a Riccati transformation and a homographie transformation to estimate both from below and above the distance between a zero and the succeeding or preceding extremum (zero of y' ) of solutions of

1.1

In this paper, we show how such transformations can be used to derive an equation from which the distance between successive zeros of a solution y of (1.1) can be estimated directly.

In this short paper we present an existence (an unicity) result for a first order differential equation in Hilbert spaces with right-hand side almost-periodic in the sense of Stepanoff.

In [1] Anderson uses a simple random sample of p-component vectors from N(μ, Σ) to test the hypothesis H, Σ = σ2I, where I is the identity matrix and σ2 is unknown. Let W= λ2/N, where λ is the likelihood ratio criterion for testing H.

In [1] Whitehead defined a CW-complex as a closure finite cell complex with the weak topology (i.e. a topology coherent with the family of closed cells, in Spanier's terminology). The purpose of this note is to show that these two conditions imposed on cell complexes can be replaced by a single one.

This note originated in an attempt to generalize the assertion of Problem 164 which was proposed by Moser [2].

Let us first state our generalization.

In 1876, Klein published the following result: “If a crunode of a real, irreducible, plane, algebraic curve changes into an acnode via the intermediary stage of a real cusp, two real inflexions are introduced in a neighborhood of the double points” [2].

Let C be an operator on the subsets of a set X with values among the subsets of X. We assume that C is a closure operator in X, i.e. a monotone, idempotent and extensive operator in X (cf., e.g., Birkhoff [3, p. 39], Schmidt [1], [2]). If A ⊆ X and B ⊆ X, we say that A and B are C-equivalent if C(A) = C(B) (Bleicher- Marczewski [4, p. 210]).

A Banach space X is said to be subspace homogeneous if for every two isomorphic closed subspaces Y and Z of X, both of infinite codimension, there is an automorphism of X (i.e. a bounded linear bijection of X) which carries Y onto Z. In [1] Lindenstrauss and Rosenthal showed that c0 is subspace homogeneous, a property also shared by l2, and conjectured that c0 and l2 are the only subspace homogeneous Banach spaces. In that paper no mention was made of subspaces of c0.

In this note we answer in the affirmative the following question.

If S is a finite family of sets such that each subset of a member of S also belongs to S, is it possible to arrange the sets in pairs so that the first member of the pair is contained in the complement of the second member?

We wish to establish the existence of a periodic solution to

1

where x, g and p are n-vectors, A is an n × n constant matrix, and ∊ is a small scalar parameter. We assume that g and p are locally Lipschitz in x and continuous and T-periodic in t, and that the origin is a point of asymptotically stable equilibrium, when ∊ = 0.