Throughout this paper we assume that every ring has unity and all modules are unital right modules. A ring R is called a (right) q-ring if every right ideal of R is quasi-injective [4], In this paper we study a generalization of this concept. A ring R is called a (right) weak q-ring (in short, wq-ring) if every right ideal of R, not isomorphic to RR, is quasi-injective. A ring R is called a right pq-ring if every proper right ideal of R is quasi-injective. Any upper triangular 2 X 2 matrix ring over a division ring is a wq-ring, which is not a q-ring. In Section 1, some general properties of wq-rings are established and, in particular, it is shown in (1.8) that a semiprime wq-ring has zero singular ideal.

Let F be a field, F* be its multiplicative group and Mn(F) be the vector space of all n-square matrices over F. Let Sn be the symmetric group acting on the set {1, 2, … , n}. If G is a subgroup of Sn and λ is a function on G with values in F, then the matrix function associated with G and X, denoted by Gλ, is defined by

and let

ℐ(G, λ) = { T : T is a linear transformation of Mn(F) to itself and Gλ(T(X)) = Gλ(X) for all X}.

When studying the category raph of finite graphs and their morphisms, Ave can exploit the fact that this category has products, [we define these ideas in detail in § 2]. This categorical product of graphs is usually called their Kronecker product, though it has been approached by various authors in various ways and under various names, including tensor product, cardinal product, conjunction and of course categorical product (see for example [6; 7; 11 ; 14; 17 and 23]).

The topic of homotopy pull-backs and push-outs has recently been discussed by a number of authors; Boardman and Vogt [5], Bousfield and Kan [6], Fantham [7], Mather [11], and Vogt [16]. Mather develops the theory with an eye to applications and of particular interest is his cube theorem which appears in this paper as Theorem (1.10); the significance of this theorem to applications is shown in [11]. As often occurs in homotopy theory the dual is not true. The purpose of this paper is to examine approximations to the dual in order to obtain new information concerning classical problems of duality.

Internal and external activities are defined for any orientation of a graph relative to a fixed labelling of its edges. It is shown that the number of such orientations of having internal activity r and external activity s is 2r+sXrs where Xrs is the coefficient of xrys in the dichromate It follows that the number of orientations of in which the resulting digraph is acyclic is given by , where is the chromatic polynomial associated with .

Consider the following commutative diagram in Top, the category of topological spaces

in which j and j' are cofibrations, p and p' are (Hurewicz) fibrations and ƒ0, ƒi and ƒ2 are homotopy equivalences.

There has been a lot of progress made on the finite-dimensional representations of species. In [3] and [11] the finite-dimensional representations of tame species are classified and in [13] it is shown that if S is a species of finite type, then every representation of 5 is a direct sum of finite-dimensional ones. However, comparatively little is known about infinite-dimensional representations.

Let A denote a complex (or real) Fréchet algebra (i.e. a complete metrizable locally m-convex algebra, see [2] or [3]). It is known [2] that the topology of such an algebra can be defined by an increasing sequence [qn] (i.e. qn(x) ≦ qn+i(x) for all x £ A and n ≧ 1) of submultiplicative (i.e. qn(xy) ≦ qn(x)qn(y) for all x, 3/ Ç ^4 and for each n ≧ I) seminorms.

A number of results are given concerning the character and cardinality of symmetrizable and related spaces. An example is given of a symmetrizable Hausdorff space containing a point that is not a regular Gδ , and a proof is given that if a point p of a symmetrizable Hausdorff space has a neighborhood base of cardinality , then p is a Gδ . It is shown that for each cardinal number m there exists a locally compact, pseudocompact, Hausdorff -space X with |X| ≧ m. Several questions of A. V. Arhangel'skii and E. Michael are partially answered.

The projectivity of the Whitehead square wn = [in, in] in π2n-1(SN) has been studied by Randall [6] who proved that if wn is projective then n must be a power of 2 or one less than a power of 2. Here we solve the question in the even case, proving by means of bo homology:

Theorem. if and only if n = 1, 2, 4.

Herrlich [7] has introduced the limit-operators to obtain every coreflective subcategory of the category Top of topological spaces and continuous maps. Using limit-operators, S. S. Hong [9] has constructed new reflective subcategories from a known extensive subcategory of a hereditary category of Hausdorff spaces and continuous maps.

Garten and Knopp [7] introduced the notion of infinite iteration of Césaro (C1 ) averages, which they called H∞ summability. Flehinger [6] (apparently unaware of [7]) produced the first nontrivial example of an H∞ summable sequence: the sequence ﹛ai ﹜ ∞i=1 where at is 1 or 0 as the lead digit of the integer i is one or not. Duran [2] has provided an elegant treatment of H∞ summability as a special case of summability with respect to an ergodic semigroup of transformations.

The literature concerning rings with polynomial identity contains several theorems in which the existence of a polynomial identity on a subring implies the existence of such an identity on the ring itself. Belluce and Jain showed in 1968 that a prime ring will satisfy a polynomial identity provided it contains a right ideal with zero left annihilator which satisfies a polynomial identity [2].

The motion groups of the real Euclidean plane and of the elliptic plane, the group of projectivities of a line, the projective general linear group PGL2(K), some orthogonal groups O3(K, Q) with char K = 2 (see [8]), are all bireflectional (zweispiegelig). There can be no doubt that bireflectional groups are of prime importance in any theory of groups that are generated by involutions. A brief look into F. Bachmann's book [1] gives convincing evidence.

Let ƒ be a complex-valued function denned on a closed subset F of the finite complex plane C, and let {Zn} be a sequence on F without limit points. We wish to find an analytic function g which simultaneously approximates ƒ uniformly on F and interpolates ƒ at the points {Zn}.

The algebraic structure of a topological algebra influences its topological structure in a way which is profound but not well understood. (See § 7 below for various examples.) Here we examine this influence rather generally, and give a fairly complete analysis of one of the many forms it can take, namely, the influence of the identities of on the group identities obeyed by the homotopy group (or groups of the components) of .

Let be a Lie group with connected Lie subgroup , and let M(t), N(i) be real analytic curves in , the Lie algebra of , with . The main result in this paper is a Lie algebraic condition which is necessary and sufficient for

In a Banach space, the directional modulus of rotundity, δ (ϵ, z), measures the minimum depth at which the midpoints of all chords of the unit ball which are parallel to z and of length at least ϵ are buried beneath the surface. A Banach space is uniformly rotund in every direction (URED) if δ (ϵ, z) is positive for every positive ϵ and every nonzero element z. This concept of directionalized uniform rotundity was introduced by Garkavi [6] to characterize those Banach spaces in which every bounded subset has at most one Čebyšev center.

A P-space is a topological space in which every Gδ-set is open. P-spaces are fairly rare. For example, the only compact (or even pseudocompact) P-spaces are finite. A larger class of spaces, the almost-P-spaces, consists of those spaces in which G δ-sets have dense interiors. The almost-P-spaces are far less restricted than the P-spaces—for example, there are infinite, compact, connected almost-P-spaces. In this paper, we study almost-P-spaces and raise a number of questions relating to them.