Let the genus of an orientable 2-manifold M be denoted by γ(M). The genus, γ(G), of a graph G is then the smallest of the numbers γ(N) for orientable 2-manifolds N in which G can be embedded. An embedding of G in M is called minimal if γ(G) = γ(M). When each component of the complement of G in M is an open 2-cell, the embedding of G in M is called a 2-cell embedding. In (3), J. W. T. Youngs has shown that each minimal embedding is a 2-cell embedding. It follows from the results of (3) that for each graph G, there is a number d(G), called the regional number, such that for any 2-cell embedding of G in an orientable 2-manifold, the number of (2-cell) complementary domains of G is ⩽d(G), with equality holding if and only if the embedding is minimal.

Properties and characterizations for prime and semiprime rings have been provided by A. W. Goldie (2, 3). In a previous paper (1), the authors used the results of (2) to characterize prime and uniform prime modules. It is the aim of the present paper to generalize Goldie's work on semi-prime rings (3) to modules. In this setting certain new properties will appear.

Notationally, in the work to follow, the symbol R always denotes a ring and all R-modules will be right R-modules.

In the theory of rings an ideal C is said to be prime if and only if whenever AB ⊆ C for ideals A and B, then either A ⊆ C or B ⊆ C. A ring is prime if the zero ideal is prime.

Let be the algebra of n X n matrices over the complex field C, X = ‖xrs‖ a matrix of , and f(X) = ‖(frs(x11, x12, … , xnn)‖ a function with domain and range in . If the frs are differentiable with respect to each of the xij on some open set R of , then the differential df(X) = ‖dfrs(xij)‖ exists, and, moreover, f(X) is Hausdorff-differentiable (HD) (1, 3, 7) i.e. df(X)is expressible in the form

where dX = ‖dxrs‖, and the matrices Ai Bi are independent of dX. The Hausdorff derivative fI(X) is defined to be

i.e. the value of df(X) for dX = I, the identity matrix (2).

Let U be the closed disk in the plane, centred at the origin, and of unit radius. By a solid packing, or briefly a packing, C of U we shall understand a sequence ﹛Dn﹜, n = 1, 2, … , of open proper disjoint subdisks of U, such that the plane Lebesgue measures of U and of are the same. If rn is the radius of Dn and the complex number cn represents its centre, then the conditions for C to be a packing are

It was proved by Mergelyan (3) that for any packing the sum of the radii diverges:

1

Mergelyan's demonstration of (1) is somewhat involved and leans heavily on the machinery of functions of a complex variable. An elegant direct proof of (1) is given by Wesler (5), who uses the technique of projecting the boundaries of the disks of the packing on a diameter I of U.

Among the unsolved problems in graphical enumeration listed in (4) is included the determination of the number of graphs and digraphs with a given partition. Parthasarathy (9) has developed a formulation for counting graphs with a given partition by making a suitable modification of the method given in (2) for the enumeration of graphs. We present here an analogous modification that leads to a formula for the number of digraphs with a given partition. Not surprisingly, the main combinatorial device for this purpose is provided by the classical theorem due to Pólya.

Singular perturbations of linear elliptic and parabolic boundary-value problems have been studied extensively by Visik and Lyusternik (7), Huet (5), and others. It is the purpose of this paper to extend the results of (5) to the non-linear elliptic and parabolic variational boundary-value problems considered during the last few years by Browder (2, 4).

In §1, we give the notations and state the main assumptions on the nonlinearity of the elliptic operators. In §2 we study the singular perturbations of non-linear elliptic variational boundary problems. In §3, we consider the case of non-linear parabolic variational boundary problems with a small parameter.

Suppose that two men stand at the same elevation on opposite sides of a mountain range and begin to climb in such a way that their elevations remain equal at all times. Will they ever meet along the way? It is this question, restated in mathematical terms, that we shall consider. We replace the mountain range by the graph of a continuous, real-valued function f(x) defined for x ∈ [0, 1], where f(0) = f(1) = 0, and we ask whether there exist continuous mappings ϕ(t), ψ(t) from [0, 1] into [0, 1] such that

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Valentine (3) introduced the three-point convexity property P3 : a set S in En satisfies P3 if for each triple of points x, y, z in S at least one of the closed segments xy, yz, xz is in S. He proved, (3 or 1) that in the plane a closed connected set satisfying P3 is the union of some three convex subsets. The problem of characterizing those sets that are the union of two convex subsets was suggested. Stamey and Marr (2) have provided an answer for compact subsets of the plane. We present here a generalization of property P3 which characterizes closed sets in an arbitrary topological linear space which are the union of two convex subsets.

Given a point set E on an open Riemann surface V we denote by H(E) the space of functions u harmonic in open sets O(u) containing E. Let V0 be a regular region of V with border α, and consider restrictions f to α of functions in H(α). For V1 = V — ⊽0, an operator L from H(α) to H(⊽1) is, by definition, normal if

1

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In (2) Halmos and Kakutani proved that any unitary operator on an infinite-dimensional Hilbert space is a product of at most four symmetries (self-adjoint unitaries). It is the purpose of this paper to show that if the unitary is an element of a properly infinite von Neumann algebra A (i.e., one with no finite non-zero central projections), then the symmetries may be chosen from A. A principal tool used in establishing this result is Theorem 1, which was proved by Murray and von Neumann (6, 3.2.3) for type II1 factors; see also (3, Lemma 5). The author would like to thank David Topping for raising the question, and for several stimulating conversations on the subject. He is also indebted to the referee for several helpful suggestions.

In the category of abelian groups the concepts of divisible group and injective group coincide. In (4) this is generalized to modules over an integral domain and it is proved for a (commutative) integral domain that the concepts of divisible module and injective module coincide if and only if the ring is hereditary if and only if the ring is a Dedekind domain.

In (8) the assumption of commutativity is dropped and the ring is assumed to have a one-sided field of quotients. The result obtained is essentially the same as in the commutative case; see the theorem following 6.2. In (13) the requirement that the ring have no zero-divisors is also dropped and the ring is assumed to possess what we have called an Ore ring (see the definition following 6.2). The result obtained is the equivalent of parts (a) and (b) of our 6.13.

Let F be a local field with ring of integers 0 and prime ideal π0. If V is a vector space over F, a lattice L in F is defined as an 0-module in the vector space V with the property that the elements of L have bounded denominators in the basis for V. If V is, in addition, a quadratic space, the lattice L then has a quadratic structure superimposed on it. Two lattices on V are then said to be isometric if there is an isometry of V that maps one onto the other.

In this paper, we consider the following problem: given two elements, v and w, of the lattice L over the regular quadratic space V, find necessary and sufficient conditions for the existence of an isometry on L that maps v onto w.

It is the purpose of this note to give some characterizations of flat and projective modules, partly in ideal theoretical terms, partly in terms of the exterior product of a module (“puissance extérieure“); cf. (1).

We shall consider left modules over a ring R with identity element and without proper zero divisors. The left module M is called flat if X ⊗R M is an exact functor on the category of right R-modules X. If M is flat over a commutative domain R, M is necessarily torsion-free. Therefore when looking for flatness of a module M over a commutative domain, one may assume from the start that M is torsion-free.

In the following theorem, we shall not restrict ourselves to commutative rings R, but the modules concerned have to be torsion-free, which, of course, should mean that rm = 0 implies r = 0 or m = 0.

Let R be a ring with a unity element. An ideal Q of R is called (right) primary if for ideals A and B of R, AB ⊂ Q and A ⊄ Q imply that Bn ⊂ Q for some positive integer n. If R satisfies the ascending chain condition for ideals (ACC), then R is said to have a Noetherian ideal theory if every ideal of R is an intersection of a finite number of primary ideals. If R is a commutative ring that satisfies the ACC, then R has a Noetherian ideal theory. However, it is known that in general R may satisfy the ACC without having a Noetherian ideal theory (an example of such a ring is given in (2)). Thus there is some interest in conditions that imply that a ring R satisfying the ACC will have a Noetherian ideal theory.

Modules are S-modules where S is an arbitrary ring with or without a unit element. We consider a projective module P having a submodule K such that K + Y = P implies that the submodule Y is P (P, then, is a projective cover of P/K (Definition 4 in this section)) and we define the submodule X of P by

Our main result states that up to isomorphism P/X is the maximal co-rational extension over P/K (by P/K, in the more precise wording of the title).

The topology of abstract 2-metric (area-metric) spaces has been the object of study in recent papers of Gähler (1) and Froda (2). The geometric properties of such spaces, however, have remained largely untouched since the initial work of Menger (3). As in ordinary metric spaces, a notion of 2-betweenness, or interiorness, can be easily defined in 2-metric spaces. In abstract metric spaces the betweenness relation is characterized among all relations defined on each triple of points of every metric space by six natural properties (4, pp. 33-40; 5). The purpose of this paper is to prove a similar theorem characterizing the relation of 2-betweenness in 2-metric spaces.

The basis theorem for directed graphs is, in effect, a result on weakly ordered sets, and, in §1, a proof is given, based on Zorn's lemma, that generalizes, and perhaps clarifies the exposition in (1, Chapter 2). In §2, a graph G* is defined, on an arbitrary collection Q of non-void subsets of a set X (which includes all its one-element subsets), in such a way that the partitions of X into Q-sets correspond to the kernels of G*. Applied to the collection Q of non-null internally stable subsets of a graph G without loops, this identifies the chromatic number of G with the least cardinal number of any kernel of G*.

A subset S of an abelian group G is said to have a centre at a if whenever x belongs to S so does 2a — x. This note is mainly concerned with self-centred sets, i.e. those S with the property that every element of S is a centre of S. Such sets occur in the study of space groups: the set of inversion centres of a space group is always self-centred. Every subgroup of G is self-centred, so is every coset in G: this is the reason why the set of points of absolute convergence of a trigonometric series is self-centred or empty (1). A self-centred set of real numbers that is either discrete or consists of rational numbers must in fact be a coset (see §3); this does not hold for an arbitrary enumerable self-centred set of real numbers (§3.3).

If γ is an infinite cardinal number, a subset S of a topological space X is said to be Pγ-embedded in X if every γ-separable continuous pseudometric on S can be extended to a γ-separable continuous pseudometric on X. (A pseudometric d on X is γ-separable if there exists a subset G of X such that |G| ⩽ 7 and such that G is dense in X relative to the pseudometric topology A pseudometric d is continuous if d is continuous relative to the product topology on X × X.) We say that S is P-embedded in X if every continuous pseudometric on S can be extended to a continuous pseudometric on X.

The general problem considered in this paper is that of sums of a finite number of reduced fractions whose numerators are elements of a finite set S of integers, and whose denominators are distinct positive integers. Egyptian, or unit, fractions are merely the case S = ﹛1﹜. Problems concerning these fractions have been treated extensively. Another specific case S = ﹛1, — 1﹜ has been treated by Sierpinski (2).