In 1954 N. Kimura proved that each idempotent in a semigroup is contained in a unique maximal subgroup of the semigroup and that distinct maximal subgroups are disjoint [13] (or see [6, pp. 21–23]). This generalized earlier results of Schwarz [14] and Wallace [15]. These maximal subgroups are important in the study of semigroups. If G is a group, then the collection S(G) of nonempty complexes of G is a semigroup and it is natural to inquire what properties of G are inherited by the maximal subgroups of S(G). There seems to be very little literature devoted to this subject. In [5, Theorem 2], with certain hypotheses placed on an idempotent, it was shown that if G is a lattice-ordered group (“1-group”) then a maximal subgroup of S(G) containing an idempotent satisfying these conditions admits a natural lattice-order. The main result of this note (Theorem 1) is that if Gis a representable 1-group and E is a normal idempotent of S(G) and a dual ideal of the lattice G, then the maximal subgroup of S(G) containing E admits a representable lattice-order.

It is proved that a pure-injective module over a commutative ring with unity is a summand of a product of duals of finitely presented modules, where duals are to be understood with reference to the circle group T, with induced module structures. Using similar techniques, it is also shown that an R-module has its underlying group pure-injective precisely when it is a submodule of a product of duals of cyclic modules and also a summand as abelian group of the same product.

All rings considered are commutative with unity and all modules are unitary. Let Mod-R be the category of modules over a ring R. An exact sequence 0 → A → B → C → 0 in Mod-R is pure-exact if, for any N in Mod-R, 0 → A⊗N → B⊗N → C⊗N → 0 is exact. A module M is pure-injective if it has the injective property relative to the class of pure-exact sequences in Mod-R. A module P is FP (finitely presented) if it is the image of a finitely generated free module with a finitely generated kernel. A module M is compact if it carries a Hausdorff compact topology so that M is a topological R-module. Let T denote the circle group—the group of real numbers modulo the integers—and let X* denote the dual module Homℤ(X, T) of the module X.

The purpose of this note is to prove a results of Jain and López-Permouth under a weaker conditions replacing R-weak injectivity by R-tightness and even getting a simpler proof.

In this paper we compute the weak Gorenstein global dimension of a coherent power series ring. It is shown that the weak Gorenstein global dimension of R[[x]] is equal to the weak Gorenstein global dimension of R plus one, provided R[[x]] is coherent.

§ 1. Introductory. The integral

where b>0, was given by Hardy (1). It was proved by applying Mellin's inversion formula. An alternative proof, based on the differential equation

satisfied by Kn(x), has been given by the author (2).

In this survey paper we present some results relating the Goldie dimension, dual Krull dimension and subdirect irreducibility in modules, torsion theories, Grothendieck categories and lattices. Our interest in studying this topic is rooted in a nice module theoretical result of Carl Faith [Commun. Algebra27 (1999), 1807–1810], characterizing Noetherian modules M by means of the finiteness of the Goldie dimension of all its quotient modules and the ACC on its subdirectly irreducible submodules. Thus, we extend his result in a dual Krull dimension setting and consider its dualization, not only in modules, but also in upper continuous modular lattices, with applications to torsion theories and Grothendieck categories.

In this note, we give a new proof of the existence of an open book decomposition for a closed non-orientable 3–manifold. This open book decomposition is analogous to a planar open book decomposition for a closed orientable 3–manifold. More precisely, in this note, we give an open book decomposition of a given closed non-orientable 3–manifold with the pages punctured Möbius bands. We also give an algorithm to determine the monodromy of this open book.

The main result of this paper is the following theorem. Let q be a prime and A be an elementary abelian group of order q3. Suppose that A acts as a coprime group of automorphisms on a profinite group G in such a manner that CG(a)′ is periodic for each a ∈ A#. Then G′ is locally finite.

We construct a norm on the nonzero elements of a Prüfer domain and extend this concept to the set of ideals of a Prüfer domain. These norms are used to study factorization properties Prüfer of domains.

Let A and C be m × m matrices and let B and D be n × n matrices, all with elements in a field F. Let AT denote the transpose of A. A well-known theorem states that, if every m × m matrix X for which AX = XA also satisfies CX = XC, then C = φ(A) for some polynomial φ(λ). In this note we establish the following simple generalizations.

Theorem 1. Let A and B have the same minimal polynomial m(λ). If each m × n matrix X over F for which AX = XB also satisfies CX = XD, then C = φ(A) and D = φ(B) for a polynomial φ(λ) over F.

The recurrence formulae for the Bessel, Legendre, hypergeometric and other such functions can all be related to each other by means of the E-functions. In this paper it will be shown how, starting from known recurrence formulae for the hypergeometric function, others can be derived. The E-function formulae are deduced in § 2, and the others in § 3.

We consider in this paper the problem (1) where Ω is the unit ball in ℝN centred at the origin, 0 ≤ α < pN,β > 0, N ≥ 3. Suppose qϵ → q as ϵ → 0+ and qϵ, q satisfy, respectively, we investigate the asymptotic estimates of the ground-state solutions (uϵ, vϵ) of (1) as β → + ∞ with p, qϵ fixed. We also show the symmetry-breaking phenomenon with α, β fixed and qϵ → q as ϵ → 0+. In addition, the ground-state solution is non-radial provided that ϵ > 0 is small or β is large enough.

If a convergent series of real or complex numbers is rearranged, the resulting series may or may not converge. There are therefore two problems which naturally arise.

(i) What is the condition on a given series for every rearrangement to converge?

(ii) What is the condition on a given method of rearrangement for it to leave unaffected the convergence of every convergent series?

The answer to (i) is well known; by a famous theorem of Riemann, the series must be absolutely convergent. The solution of (ii) is perhaps not so familiar, although it has been given by various authors, including R. Rado [7], F. W. Levi [6] and R. P. Agnew [2]. It is also given as an exercise by N. Bourbaki ([4], Chap. III, § 4, exs. 7 and 8).

The theory of algebraic curves associated with subgroups of finite index in the modular group Γ is highly developed for such subgroups of Γ as may be defined by means of congruences in the ring ℤ of rational integers. The situation in he case of non-congruence subgroups of Γ, on the other hand, is drastically different. It reduces to a few isolated examples, two of which may be found in [12]. Related research by A. O. L. Atkin and H. P. F. Swinnerton-Dyer began in [1].

In this note we prove that the equation x2 + 1 = yn, x, y, n ɛ ℕ, n>2, has no solutions (x, y, n)with 2 × y. Moreover, all solutions (x, y, n)of the equation with 2| y satisfy n < 5. 106 and y < exp exp exp 30.

In this paper, by using the Leggett–Williams fixed point theorem, the existence of three positive periodic solutions for differential equations with piecewise constant argument and impulse on time scales is investigated. Some easily verifiable sufficient criteria are established. Finally, an example is given to illustrate the results.

In Section 33 of [2], Bonsall and Duncan define an element t of a Banach algebra to act compactly on if the map a → tat is a compact operator on . In this paper, the arguments and technique of [1] are used to study this question for C*-algebras (see also [10]). We determine the elements b of a C*-algebra for which the maps a → ba, a → ab, a → ab + ba, a → bab are compact (respectively weakly compact), determine the C*-algebras which are compact in the sense of Definition 9, of [2, p. 177] and give a characterization of the C*-automorphisms of which are weakly compact perturbations of the identity.

The KO-cohomology ring of the symmetric space SU(2n)/SO(2n) is computed by using the Bott exact sequence and some facts on the real and quaternionic representation rings of SU(2n) and SO(2n).

Let $\sigma _q \,:\,{{\mathbb{R}}^q} \to{\textbf{S}}^q\setminus N_q$ be the inverse of the stereographic projection with center the north pole $N_q$. Let $W_i$ be a closed subset of ${\mathbb{R}}^{q_i}$, for $i=1,2$. Let $\Phi \,:\,W_1 \to W_2$ be a bi-Lipschitz homeomorphism. The main result states that the homeomorphism $\sigma _{q_2}\circ \Phi \circ \sigma _{q_1}^{-1}$ is a bi-Lipschitz homeomorphism, extending bi-Lipschitz-ly at $N_{q_1}$ with value $N_{q_2}$ whenever $W_1$ is unbounded.

As two straightforward applications in the polynomially bounded o-minimal context over the real numbers, we obtain for free a version at infinity of: (1) Sampaio’s tangent cone result and (2) links preserving re-parametrization of definable bi-Lipschitz homeomorphisms of Valette.

A long-standing problem is the characterization of subsets of the range of a vector measure. It is known that the range of a countably additive vector measure is relatively weakly compact and, in addition, possesses several interesting properties (see [2]). In [6] it is proved that if m: Σ → Χ is a countably additive vector measure, then the range of m has not only the Banach–Saks property, but even the alternate Banach-Saks property. A tantalizing conjecture, which we shall disprove in this article, is that the range of m has to have, for some p > 1, the p-Banach–Saks property. Another conjecture, which has been around for some time (see [2]) and is also disproved in this paper, is that weakly null sequences in the range of a vector measure admit weakly-2-summable sub-sequences. In fact, we shall show a weakly null sequence in the range of a countably additive vector measure having, for every p < ∞, no weakly-p-summable sub-sequences.