In this note it is proved that a (real or complex) semiprime Banach algebra A satisfying xAx = x2Ax2 for every x ∈ A is a direct sum of a finite number of division Banach algebras.

Let f be a continuous function, and u a continuous linear function, from a Banach space into an ordered Banach space, such that f − u satisfies a Lipschitz condition and u satisfies an inequality implicit-function condition. Then f also satisfles an inequality implicit-function condition. This extends some results of Flett, Craven and S. M. Robinson.

For a wide class of sine trigonometric series we obtain an estimate for the integral modulus of continuity.

We present two more characterizations of maps which preserve orthogonal decompositions defined on Hilbert spaces ordered by natural cones.

If a property P of ring elements satisfies conditions (a)-(d), then the largest homomorphically closed class having no non-zero P-element is a dual radical in the sense of Andrunakievič, and every dual radical can be obtained in this way. Also properties defined by polynomials are considered and as an application we get various characterizations of the Behrens radical.

A method is developed for obtaining compact, easily computed and statistically interpretable expressions for the generalized k-statistics associated with multiply-indexed arrays of random variables such as those which arise in variance component analysis. These expressions will be used in the next paper in this series to give formulae for variances and covariances of estimates of components of variance.

Let P be the probability distribution of a sample without replacement of size n from a finite population represented by the set N={1,2,…N}. For each r=0, 1, …, an approximation Pr is described such that the uniform norm ‖P − Pr‖ is of order (n2/N)r+1 if n2/N→0. The approximation Pr is a linear combination of uniform probability product-measures concentrated on certain subspaces of the sample space Nn.

Topologise the set of continuous self-mappings of a Hausdorff space by the graph topology. When the set of closed subsets of the space is given the upper semi-finite topology then the function which assigns to a map its fixed-point set is continuous. In many familiar cases this is the largest such topology. Related results also hold for the function which assigns to each pair of maps their coincidence set.

We study the embeddings of a finite p-group U into Sylow p-subgroups of Sym (U) induced by the right regular representation p: U→ Sym(U). It turns out that there is a one-to-one correspondence between the chief series in U and the Sylow p-subgroups of Sym (U) containing Up. Here, the Sylow p-subgroup Pσ of Sym (U) correspoding to the chief series σ in U is characterized by the property that the intersections of Up with the terms of any chief series in Pσ form σp. Moreover, we see that p: U→ Pσ are precisely the kinds of embeddings used in a previous paper to construct the non-trivial countable algebraically closed locally finite p-groups as direct limits of finite p-groups.

Based on a Junnila's paracompactness characterization we give a definition of pairwise paracompact space which permits us to prove that a bitopological space is quasi-metrizable if, and only if, it is a pairwise developable and pairwise paracompact space. An easy consequence of this result is the biquasi-metric form of the Morita metrization theorem. We also give some results on open mappings and strong quasi-metrics.

A module is uniserial if its lattice of submodules is linearly ordered, and a ring R is left serial if R is a direct sum of uniserial left ideals. The following problem is considered. Suppose the injective hull of each simple left R-module is uniserial. When does this imply that the indecomposable injective left R-modules are uniserial? An affirmative answer is known when R is commutative and when R is Artinian. The following result is proved.

Let R be a left serial ring and suppose that for each primitive idempotent e, eRe has indecomposable injective left modules uniserial. The following conditions are equivalent. (a) The injective hull of each simple left R-module is uniserial. (b) Every indecomposable injective left R-module is univerial. (c) Every finitely generated left R-module is serial.

The rest of the paper is devoted to a study of some non-Artinian serial rings which serve to illustrate this theorem.

We prove that every (locally) contractible topological group is (L)EC and apply these results to homeomorphism groups, free topological groups, reduced products and symmetric products. Our main results are: The free topological group of a θ-contractible space is equiconnected. A paracompact and weakly locally contractible space is locally equiconnected if and only if it has a local mixer. There exist compact metric contractible spaces X whose reduced (symmetric) products are not retracts of the Graev free topological groups F(X) (A(X)) (thus correcting results we published ibidem).

Let R, S be a partition of 2, 3,… so that rational powers fall in the same class. Let (λn) be any real sequence; we show that there exists a set N, of dimension 1, so that (x + λn) (n = 1,2, …) are normal to every base from R and to no base from S, for every x ∈ N.

Consider a compact zero dimensional (profinite) monoid. While the group of units must be open, a regular D-class need not be open in the ideal it generates. This is the case if and only if the semigroup contains infinitely many copies of a certain semilattice composed of an increasing sequence of idempotents converging to an upper bound.

Using compactifications of free products, two generator compact monoids with these properties are constructed.

In this paper we prove that a pairwise Hausdorff bitopological space is quasi-metrizable if and only if for each point x ∈ X and for i, j = 1,2, i ≠ j, one can assign nbd bases { S(n, i; x) | n = 1, 2,… } such that (i) y ∉ S (n − 1, i; x) imples S(n, i; x) ∩ S (n, j; y) = φ, (ii) y ∈ S (n, i; x) implies S (n, i; y) ⊂ S(n − 1, i; x). We derive two further results from this.

Let G be a group factorized by finitely many pairwise permutable nilpotent subgroups. The aim of this paper is to find conditions under which at least one of the factors is contained in a proper normal subgroup of G.