This paper is concerned with the internal structure of Ext(Q, T) where Q is the group of rationals and T a reduced p-primary group of unbounded order. In [1] Irwin, Khabbaz, and Rayna define the splitting length of an arbitrary abelian group A, written l(A), to be the least positive integer n, otherwise infinity, such that A ⊗ . . . ⊗ A (n factors) splits. The concept of splitting length has been induced on Ext(Q, T), see [2; 5].

Let be a complex Banach space and the algebra of bounded operators on . M. Rosenblum's theorem [13; 12] (also discovered by M. G. Kreĭn, cf. [9]) states that (if A, B are fixed bounded operators) the spectrum of the operator on defined by = AX – XB is contained in σ (A) – σ(B) = {α – β : α∊σ(A), β∊σ(B)}. In particular, the condition σ(A) ∩ σ(B) = Ø implies that for each Y ∊ there is a unique X ∊ such that AX – XB = Y. This does not completely settle the question of solvability of the equation AX – XB = Y: for example, if A is the backward unilateral shift and B = 0, then the equation has a solution (for any Y) even though σ(B) ⊆ σ(A).

Let F be the group freely generated by the countably infinite set X = {x1, x2, . . . ,xi, . . . }. Let w(x1, x2, . . . , xn) be a reduced word representing an element of F and let G be an arbitrary group. Then V(w, G) will denote the set

whose elements will be called values of w in G. The subgroup of G generated by V(w, G) will be called the verbal subgroup of G with respect to w and be denoted by w(G).

Let-R be a commutative ring with 1, and let

be the symmetric algebra of an R-module M. We regard the isomorphisms S0(M) ≅ R and S1(M) ≅ M a s identifications. There is a unique R-algebra homomorphism Δ : S(M) → S(M) ⊗RS(M) (called the comultiplication) satisfying Δ(m) = m ⊗ 1 + 1 ⊗ m for all m ∊ M; any element x ∊ S(M) for which Δ(x) = x ⊗ 1 + 1 ⊗ x is said to be primitive. The set of all primitive elements in S(M) is denoted P(M).

Utilizing a matrix representation of semiperfect rings by a family of bimodules over local rings, we describe the structure of generalized quasi-Frobenius rings in two steps: a cyclic generalized quasi-Frobenius ring is a matrix ring over a cycle of Morita dualities between local rings, and an arbitrary generalized quasi-Frobenius ring is a matrix ring over a family of cyclic generalized quasi-Frobenius rings.

Our results provide a complete classification of generalized quasi-Frobenius rings, modulo the classification of local rings with Morita duality, of certain bimodules over such rings, and of certain rest families of multiplication maps.

Let k be a real closed field (see § 2 for a definition). Let be an algebraic closure of k. An algebraic set denned over k is, as usual, a subset of (n some integer greater than 0) which is the set of zeros of some polynomials in k[X1, . . . , Xn]. A variety is denned to be an absolutely irreducible algebraic set. We define the real points of an algebraic set X to be the points in X ∩ kn. One can then define X to be real if I(X ∩ kn) = I(X). (I(X) = the polynomials in k[X1, . . . , Xn] which vanish on X.)

There has been considerable interest recently in the investigation of "Korovkin sets". Briefly, for X a Banach space and a family of linear operators on X, a subset K ⊂ X is a Korovkin set relative to if for any bounded sequence {Tn} ⊂ , Tnk → k in X for each k ∊ K implies Tnx → x for each x ∊ X. A large portion of these investigations have been carried out for X being one of the spaces C(S), S compact Hausdorff, the usual Lp spaces of functions on some finite measure space, or some Banach lattice; while is one of the classes +-positive operators, 1-contractions (i.e., ||T|| 1), or + ⋂1

Every completely regular space has at least one Hausdorff compactification and much research in Topology has been devoted to methods of constructing the compactifications of completely regular spaces. These methods fall into two general categories: internal methods and external methods. External methods are characterized by their reliance on structures which are not topological or outside the immediate topological structure of the space in question; examples of the former are A. Weil [20] - uniform structures and Yu. Smirnov [15] who uses the proximity structures of Efremovich; using the continuous real-valued functions to produce an embedding of the space serves as an example of the latter.

A space X is called countdbly metacompact (countably paracompact) if every countable open cover has a point finite (locally finite) open refinement. According to Hodel [5], a space X is called countably subparacompact if every countable open cover has a σ-discrete closed refinement. It is well-known (see Mansfield [10] and Dowker [4]) that in normal spaces all of the preceding notions are equivalent. Also, according to Hodel [5], a countably subparacompact space is countably metacompact and the reverse implication is false.

Let C(I) denote the linear space of continuous functions from the compact interval I = [a, b] into n-dimensional real arithmetic space Rn, and let C′(I) be the subspace of continuously differentiable functions on I. A general boundary value problem for a first-order system of n ordinary differential equations on I is given by

Let be a fixed separable, infinite dimensional complex Hilbert space, and let () denote the algebra of all (bounded, linear) operators on . The ideal of all compact operators on will be denoted by and the canonical quotient map from () onto the Calkin algebra ()/ will be denoted by π.

Some open problems in the theory of extensions of C*-algebras (cf. [1]) have recently motivated an increasing interest in the class of all operators in () whose self-commuta tor is compact.

Let A be a separable C*-algebra. Two representations π and π1 of A on the Hilbert spaces H and H1, respectively are said to be quasi-equivalent (denoted by π ~ π1) if projections of H ⊕ H1 on the invariant subspaces H and H1 of (π ⊕ π1)(A) have the same central support in the commutant (π ⊕ π1) (A)′ of (π ⊕ π1) (A), or equivalently, if there is an isomorphism ϕ of π(A)″ onto π1(A)″ such that ϕ(π(x)) = π(x) for all x ∊ A (cf. [5, § 5]). A representation π of A is said to be a factor representation if the center of π(A)″ consists of scalar multiples of the identity.

The notion of almost convergence introduced by Lorentz [15] has been generalized in several directions (see, for example [1; 8; 11 ; 14; 17]). I t is the purpose of this paper to give a generalization based on the original definition in terms of invariant means. This is effected by replacing the shift transformation by an "ergodic" semigroup of positive regular matrices in the definition of invariant mean. The resulting "- invariant means" give rise to a summability method which we dub -almost convergence.

In I we obtained an equation, called the "chromatic equation", for the generating function g(x, y, z, λ). In II and III we obtained special equations, valid in the cases λ = τ + 1 and λ = 3 respectively, for the generating function l(y, z, λ), defined as the coefficient of x2 in g(x, y, z, λ). The argument was independent of that in I and no attempt was made to derive the new formulae from the chromatic equation.

The intent of this paper is to show that the additive l-group of an f-ring S determines the ring structure. This is why there are so many papers that simply extend known results for abelian l-groups to f-rings. Theorem 3.1 asserts that there is a one-to-one correspondence between the f-multiplications on S and a set of homomorphisms from the positive cone of the l-group S into the positive cone of the ring (S) of polar preserving endomorphisms of the l-group S. In fact, each f-multiplication of S is determined by a homomorphism of S+ into (S)+.

In [1], S. S. Chern gave a very elegant and simple proof of the Gauss-Bonnet formula for closed (i.e. compact without boundary) oriented Riemannian manifolds of even dimension:

Here, c is a suitable constant depending on the dimension of M and Ω is an n-form (n = dim M) which may be calculated from its curvature tensor. W. Greub gave a coordinate-free description of this integrand Ω (cf. [4]).

The aim of this paper is to find necessary and sufficient conditions on a group G and a ring A for the group ring AG to be semi-perfect. A complete answer is given in the commutative case, in terms of the polynomial ring A[X] (Theorem 5.8). In the general case examples are given which indicate a very strong interaction between the properties of A and those of G. Partial answers to the question are given in Theorem 3.2, Proposition 4.2 and Corollary 4.3.

Let Λ be a ring and σ a kernel functor (left exact preradical) on the category of left Λ-modules. A left Λ-module M is called σ-semi-simple if whenever N is a submodule of M with M/Nσ-torsion, N is a direct summand of M. In Section 1 we consider alternative characterizations and properties of σ-semi-simplicity for modules. In Section 2 conditions equivalent to the σ-semi-simplicity of the ring are obtained. Section 3 is devoted to the condition, which frequently arises in Section 2, that every σ-torsion module be semisimple.

A theorem of Marshall Osborn [15] states that a simple ring with involution of characteristic not 2 in which every non-zero symmetric element is invertible must be a division ring or the 2 × 2 matrices over a field. This result has been generalized in several directions. If R is semi-simple and every symmetric element (or skew, or trace) is invertible or nilpotent, then R must be a division ring, the 2 × 2 matrices over a field, or the direct sum of a division ring and its opposite [6; 8; 13; 16].

Throughout, we consider a finite Galois extension L|K of non-archimedian valued fields which are maximally complete [2, Chapter 2], Let v denote the valuation on L and let L* denote the group of non-zero elements of L. We mayidentify the value group v(L*) of L with a subgroup of D, where D denotes the minimal divisible ordered group containing v(K*). We denote the residue field of L by , and will always assume that the field extension is separable. The characteristic of will invariably be denoted by p ; much of what follows is trivial in case p = 0.