Throughout this paper the word “ring” will mean an associative ring which need not have an identity element. There are Artinian rings which are not Noetherian, for example C(p∞) with zero multiplication. These are the only such rings in that an Artinian ring R is Noetherian if and only if R contains no subgroups of type C(p∞) [1, p. 285]. However, a certain class of Artinian rings is Noetherian. A famous theorem of C. Hopkins states that an Artinian ring with an identity element is Noetherian [3, p. 69]. The proofs of these theorems involve the method of “factoring through the nilpotent Jacobson radical of the ring”. In this paper we state necessary and sufficient conditions for an Artinian ring (and an Artinian module) to be Noetherian. Our proof avoids the concept of the Jacobson radical and depends primarily upon the concept of the length of a composition series. As a corollary we obtain the result of Hopkins.

The two-dimensional unitary space, U2, is a complex vector space of points (x, y) = (x1 + ix2, y1 + iy2), for which the distance between (x, y) and (x', y') is defined by . A unitary transformation is a linear transformation which preserves distance. A line is the set of points (x, y) satisfying some complex equation ax + by = c. A unitary transformation is a (unitary) reflection if it is of finite period n > 1 and leaves a line pointwise invariant. Thus à unitary matrix represents a reflection if its two characteristic roots are 1 and a complex nth root (n > 1) of 1.

The purpose of this paper is to establish the following theorems, which were obtained by Hopf and Voss in their joint paper (2) for the case where n = 2.

THEOREM 1. Let Vn, V*n be two closed orientable hyper surfaces twice differ entiably imbedded in a Euclidean space En+l of dimension n + 1 ≥ 3.

In (3), Dekker drew attention to an analogy between (a) the relationship of the recursive sets to the recursively enumerable sets, and (b) the relationship of the retraceable sets to the regressive sets. As was to be expected, this analogy limps in some respects. For example, if a number set α is split by a recursive set, then it is decomposed by a pair of recursively enumerable sets; whereas, as we showed in (6, Theorem 2), α may be split by a retraceable set and yet not decomposable (in a liberal sense of the latter term) by a pair of regressive sets. The result for recursive and recursively enumerable sets, of course, follows from the trivial fact that the complement of a recursive set is recursive.

Let be the class of m × n matrices all of whose entries are either 0 or 1 where every matrix A in the class satisfies the conditions that row i of A has ri ones, i = 1, 2, . . . , m; and column j of A has sj ones, j = 1, 2, . . . , n. We let R = (r1 . . . , rm), S = (s1, . . . , sn), and assume that r1 ≥ r2 ≥ . . . ≥ rm ≥ 0; s1 ≥ s2 ≥ . . . ≥ sn > 0. When this is the case we say the class is normalized.

In this paper we develop a theory of maximal and minimal operators and their duals associated with the system

Let R be a commutative ring with identity 1. A character of an R-algebra E is a homomorphism from E onto R, regarded as an algebra over itself. If (Eα) is a family of R-algebras indexed by a set A and if

then for every β ∈ A and every character vβ of Eβ, vβ o prβ is a character of E where prβ is the projection homomorphism from E onto Eβ. Further if A is finite and if the only idempotents of R are 0 and 1 (equivalently, if R is not the direct sum of two proper ideals), it is easy to see that every character of E is of this form.

We call a subset X of a group an elliptic set if there is an integer n such that each element of the group generated by X can be written as a product of at most n elements of X ∪ X−1. The terminology is due to Philip Hall, who investigated elliptic sets in lectures given in Cambridge in the 1960's. Hall was chiefly interested in sets X which are unions of conjugacy classes, but among other things he proved that if H, K are subgroups of a finitely generated nilpotent group then their union H ∪ K is elliptic. We shall say that a subgroup H of an arbitrary group G is elliptically embedded in G, and we write H ee G, if H ∪ K is an elliptic set for each subgroup K of G. Thus H ee G if for each subgroup K there is an integer n (depending on K) such that

where the product has 2n factors.

Let $X$ be a smooth projective surface over the complex number field and let $L$ be a nef-big divisor on $X$. Here we consider the following conjecture; If the Kodaira dimension $\kappa (X)\ge 0$, then ${{K}_{X}}L\,\ge \,2q(X)\,-\,4$, where $q\left( X \right)$ is the irregularity of $X$. In this paper, we prove that this conjecture is true if (1) the case in which $\kappa (X)=0$ or 1, (2) the case in which $\kappa (X)=2$ and ${{h}^{0}}(L)\,\ge \,2$ , or (3) the case in which $\kappa (X)=2$, $X$ is minimal, ${{h}^{0}}(L)\,=\,1$ , and $L$ satisfies some conditions.

We shall give a simple characterization of a lattice homomorphism from a linear lattice E to a linear lattice F. This paper is motivated by the following two theorems in Kaplan [2] :

If ϕ is a lattice homomorphism, then ϕt(Fb) is an ideal in Eb.

(2) If ϕ is a lattice homomorphism, then ϕtt is a lattice homomorphism from ϕbb into ϕbb.

The main theorem is stated and proved in section 3. In section 1, we shall give notations and in section 2, we shall prove a main lemma. For details, we refer to Vulikh [3].

The object of this paper is to prove the following theorem, a special case of which was previously explored in [1].

THEOREM. Let R be any associative ring with the property that

(†) for each x,y ∊ R, there exist integers m,n ≧ I for which xy = ymxn.

Let G be an abelian group, written additively. A complexvalued function ƒ, defined on G, is said to be positive definite if the inequality

1

holds for every choice of complex numbers C1, …, cn and S1, …, sn in G. It follows directly from (1) that every positive definite function is bounded. Weil (9, p. 122) and Raïkov (5) proved that every continuous positive definite function on a locally compact abelian group is the Fourier-Stieltjes transform of a bounded positive measure, thus generalizing theorems of Herglotz (4) (G = Z, the integers) and Bochner (1) (G = R, the real numbers).

If ƒ is a continuous function, then condition (1) is equivalent to the condition that

2

The present work is a sequel to our previous article (10) with the same title. The major theme remains the study of the relationship between tensor products of spaces of functions and vector-valued measures on a space S. In (10) S was a compact Hausdorff space. Here we extend our considerations to locally compact Hausdorff spaces. B(S) still stands for the Borel class of S.

Three types of vector-valued measures m : B(5) → E, E a Banach space, are considered here (§3), namely, weak*, weak, and strong vector measures, but the concepts of weak and strong measures coincide. This result is due to Bartle, Dunford, and Schwartz (1) for abstract vector measures, i.e. defined on an abstract σ-algebra of sets.

The convex hull of an arbitrary set M in real Euclidean n-space is known to consist of all the points within the r-simplexes with r + 1 vertices from M, r ≤ n. This note shows that if M is specialized to be a curve An of real order n, then its convex hull consists of all the points within the r-simplexes with r + 1 vertices on An, n = 2r + 1 or n = 2r.

Titchmarsh (4) has shown how the classical method of complex variables can be used to obtain expansion theorems for the singular cases of the second order equation

(1) .

The purpose of this paper is to indicate how these results can be generalized to the singular cases of the pair of first order equations

,

Let T denote a closed unit equilateral triangle. For a fixed integer n, let dn denote the infimum of all those x for which it is possible to partition T into n subsets, each subset having a diameter not exceeding x. We recall that the diameter of a plane set A is given by

where ρ (a, b) is the Euclidean distance between a and b.

In this note we determined dn for some small values of n. Typical values of dn are given in Table I. These values were obtained by three methods. As would be expected, as the value of n increases, the complexity of the argument needed to obtain dn also increases. We begin with the simplest case.

If f ∈ Lp (— ∞, ∞), 1 < p ≤ 2, then f has a Fourier-Plancherel transform F ∈ Lq (— ∞, ∞) where p-1 + q-1 = 1. Also if ∣x∣1-2/qf(x) ∈ Lq (— ∞, ∞), q ≥ 2, then / has a Fourier-Plancherel transform F ∈ Lq (— ∞, ∞). These results can be found in (2, Theorems 74 and 79). In neither case, however, does the collection of transforms cover Lq, except when p = q = 2, and in neither case, with the same exception, has the collection of transforms been characterized.

Further, if f ∈ Lp, (— ∞, ∞), 1 < p ≤ 2, then its transform F has the property |x|1-2/pF(x) ∈ Lp (— ∞, ∞) (see 2, Theorem 80) but, except when p = 2, the collection of transforms does not cover the set of functions with this property, and again, except when p = 2, the collection of transforms has not been characterized.

Let f(z) be a non-constant entire function and let {dn} be a sequence of complex numbers such that

The set of equations

(1.1)

The method of A. Young for finding the set of primitive idempotents of the group algebra of the symmetric group is classical; it was first given by Frobenius (4) using results of Young (10 and 11). A concise account can be found in (9) and a very detailed treatment in (6).

In (1), M. Davis and M. Maschler define the kernel K of a characteristic-function game; they also prove, among other theorems, that K is a subset of the bargaining set M1(i) and that it is never void, i.e. that for each coalition structure b there exists a payoff vector x such that the payoff configuration (x, b) belongs to K. The main advantage of the kernel, as it seems to us, is that it is easier to compute in many cases than the bargaining set M1(i).