R denotes an associative ring with identity. Module means unitary right R-module. A module has finite Goldie dimension over R if it does not contain an infinite direct sum of nonzero submodules [6]. We say R has finite (right) dimension if it has finite dimension as a right R-module. We denote the fact that M has finite dimension by dim (M)<∞.

A nonzero submodule N of a module M is large in M if N has nontrivial intersection with nonzero submodules of M [7]. In this case M is called an essential extension of N. N⊆′M will denote N is essential (large) in M. If N has no proper essential extension in M, then N is closed in M. An injective essential extension of M, denoted I(M), is called the injective hull of M.

In [7], Goebel, Kirk and Shimi proved the following:

Theorem. Let X be a uniformly convex Banach space, K a nonempty bounded closed and convex subset of X, and F:K→K a continuous mapping satisfying for each x, y∈K:

(1)

where a i ≥0 and Then F has a fixed point in K.

In this paper we shall prove that this theorem remains true in any Banach space X, provided that K is a nonempty, weakly compact convex subset of X and has normal structure (see Definition 1 below).

If E is a topological space, then according to Mrowka in [7], a space X is E-incompletely regular if X can be embedded as a subspace of a topological product of copies of E, and X is E-compact if X can be embedded as a closed subspace of a product of copies of E. The following is [7, Theorem 4.14].

In this paper we consider an irreducible *-regular ring with order k for some k≥4. If is also a Baer ring it is a rank ring. Our first result is:

Theorem 1.3. Let be an irreducible *-regular Baer ring with order k for some k≥4. The following are equivalent.

Constructions are made of a T 1 space which does not have a T 1 completion and of a quasi-uniform space which is complete, but not strongly complete. An example relating to a completion due to Popa is given. An alternate definition for Cauchy filter, called C-filter, is examined and a construction of a C-completion is given. We discuss quasi-pseudometrics over a Tikhonov semifield R Δ. Every topological space is quasi-pseudometrizable over a suitable R Δ. It is shown that if a quasi-pseudometric space over R Δ is complete, the corresponding quasi-uniform structure is C-complete. A general method for constructing compatible quasiuniform structures is given.

Let (X, τ) be a Tychonov space and the collection of all families of pseudometrics on X generating the topology τ on X. Let f:X→X and c>0. Then f is said to be a topological c-homothety if there exists some B in such that d(f(x), f(y))=cd(x, y) for all d ∈ B and all x, y in X (see [4]). We say that f can be linearized in L as a c-homothety if there exists a linear topological space L, and a topological embedding i:X→L such that i(f(x))=c i(x) for all x in X (see [4]).f is said to be squeezing if for some a in X.

Let R be a ring such that, for each element a of R, there exists a positive integer n(a) > 1, depending on a, such that a n(a)=a. Jacobson proved that such a ring R is necessarily commutative and the purpose of this paper is to give a proof of Jacobson’s Theorem that does not involve the use of the axiom of choice.

We take this opportunity to point out that all rings are assumed to be associative and that nothing beyond elementary ring theory is assumed in the proof to follow. Such ring theory can be found, for example, in [1].

In this note we compute the derived functors of “torsion submodule” using a certain duality result for semi-exact functors on an abelian category of finite global dimension.

Let F be a field, G a subgroup of S m , the symmetric group of degree m, and χ a linear character on G, i.e., a homomorphism of G into the multiplicative group of F. Let V 1,...,V m be vector spaces over F such that Vi = V σ(i) for i=1,…,m and for all σ∈G. If W is a vector space over F, then a m-multilinear function is said to be symmetric with respect to G and χ if

for any σ ∊ G and for arbitrary xi ∊ Vi.

In the latter part of the 1950’s some interesting papers appeared (e.g. [2] and [10]) which examined the relationships occurring between the purely algebraic and homological aspects of the theory of finitely generated modules over Noetherian rings. Many of these relationships remain valid if one considers the much wider class of rings determined by the following definition.

Definition. A commutative ring R is called pseudo-Noetherian if it satisfies the following two conditions.

Gurland [1] by making use of the Cramer-Rao lower bound for the variance of an unbiased estimator obtained the following inequality

1

for real values of α and δ satisfying α+δ>0, α≠0, δ>0. He used the fact that (Γ(δ)Γ(δ+α))x α is an unbiased estimator of θ α where θ is the parameter for the density function

Let A be an n × n complex matrix. The singular values of A are the non-negative square-roots of the eigenvalues of A*A. G. N. De Oliviera [4] gave a necessary condition for the existence of a matrix A with a 1..., a n as diagonal elements and α1,..., αn as singular values. We shall give another necessary condition which implies the above author’s condition and we show that this is also a sufficient condition for the case n =2.

An element a of a semigroup S is n-potent if there exist a 1, a 2,..., a k ∈S such that a = a 1 a 2...a k and If S is a group, the set of n-potent elements is a normal subgroup of S and the set of 1-potent elements is the commutator subgroup of S.

There are few known metrization theorems for manifolds (locally Euclidean, connected, Hausdorff space). It is well known that for manifolds metrizability, second countability, Lindelöf's condition, σ-compactness and paracompactness are equivalent. Although these conditions imply separability, the latter does not imply any of the former (see Example 2.2), as is often believed. A common source of metrization for a covering manifold is that lifted from the base manifold [8; p. 181].

For 2-manifolds, the presence of a complex analytic structure gives us a metrization theorem; it has been shown [1] that such manifolds are topologically characterized as those which are orientable and second countable.

As observed by A. Bielecki and others ([1], [3]) the Banach contraction principle, when applied to the theory of differential equations, provides proofs of existence and uniqueness of solutions only in a local sense. S. C. Chu and J. B. Diaz ([2]) have found that the contraction principle can be applied to operator or functional equations and even partial differential equations if the metric of the underlying function space is suitably changed.

Let {s m} be a discrete set of points in ℂN and λm any sequence of points in ℂ. We shall be interested in finding an entire function F(z) such that F(s m )=λm . This is of course easy if no restriction is placed on F, but we shall be interested in finding an F which in addition satisfies certain growth conditions.

We shall denote the variable z=(z 1,..., z N), z j =x j +iy j ,

Let R be a prime ring and T be a nontrivial automorphism of R. If xxT —xTx is in the center of the ring for every x in R, then R is a commutative integral domain.

Recently there has been a great deal of interest in extending refinements of locally finite and point finite collections on subsets of certain topological spaces. In particular the first named author showed that a subset S of a topological space X is P-embedded in X if and only if every locally finite cozero-set cover on S has a refinement that can be extended to a locally finite cozero-set cover of X. Since then many authors have studied similar types of embeddings (see [1], [2], [3], [4], [6], [8], [9], [10], [11], and [12]). Since the above characterization of P-embedding is equivalent to extending continuous pseudometrics from the subspace S up to the whole space X, it is natural to wonder when can a locally finite or a point finite open or cozero-set cover on S be extended to a locally finite or point-finite open or cozero-set cover on X.