Let n∈ℕ and let Fn be the free group on n generators. Let w be an arbitrary word in Fn, and let σ be an n-cycle in Sn. We consider groups of the type Γ(n,w)=Fn/N, where N is the normal closure in Fn of the “cycled words’’ w, σ(w), σ2(w),…,σn−1(w), and solve, by means of classical algebraic number theory, the following problems.

A. When is Γ(n,w)ab infinite?

B. When is Γ(n,w) a perfect group?

Rings in which each right ideal is quasi-continuous (right π-rings) are shown to be a direct sum of semisimple artinian square full ring and a right square free ring. Among other results it is also shown that (i) a nonlocal right continuous indecomposable right π-ring is either simple artinian or a ring of matrices of a certain type, and (ii) an indecomposable non-local right continuous ring is both a right and a left π-ring if and only if it is a right q-ring. In particular, a non local indecomposable right q-ring is a left q-ring.

In a recent illuminating paper, June M. Parker [5] discussed Choquet integral representations of comonotonic additive functionals and related concepts. In our paper we provide a generalization of the Choquet integral and use this to obtain an integral representation for comonotonic additive operators.

The primary purpose of this paper is to provide general sufficient conditions for any real quadratic order to have a cyclic subgroup of order n∈ℕ in its ideal class group. This generalizes results in the literature, including some seminal classical works. This is done with a simpler approach via the interplay between the maximal order and the non-maximal orders, using the underlying infrastructure via the continued fraction algorithm. Numerous examples and a concluding criterion for non-trivial class numbers are also provided. The latter links class number one criteria with new prime-producing quadratic polynomials.

An example of a non-topologizable algebra was given in [2]. In [4] Żelazko gave a simple proof of the fact that, if X is an infinite-dimensional vector space, then the algebra of all finite-rank linear operators on X is not topologizableas a topological algebra. In the following we use a similar idea to prove that, if E is a Fréchet space which is not normable, then each subalgebra A of the algebra of all bounded linear operators on E such that A contains the ideal of continuous, finite-rank operators, is non-topologizable as a topological algebra. This is a shorter proof and more general version of the result of [1].

Let S be an (ideal) extension of a completely 0-simple semigroup S0 by a completely 0-simple semigroup S1. Congruences on S can be uniquely represented in terms of congruences on S0 and S1. In this representation, for a congruence ρ on S, we express ρK,ρT,ρK and ρT, where these denote the least (greatest) congruences with the same kernel (trace) as ρ. Let κ be the least completely 0-simple congruence on S. We provide necessary and sufficient conditions, in terms of the kernel of κ, in order that the relation K be a congruence, and also that [Cscr](S)/K be a modular lattice, where [Cscr](S) denotes the congruence lattice of S.

We show that if R is a ring such that each minimal left ideal is essential in a (direct) summand of RR, then the dual of each simple right R-module is simple if and only if R is semiperfect with Soc(RR)=Soc(RR) and Soc(Re) is simple and essential for every local idempotent e of R. We also show that R is left CS and right Kasch if and only if R is a semiperfect left continuous ring with Soc(RR)⊆eRR. As a particular case of both results we obtain that R is a ring such that every (essential) closure of a minimal left ideal is summand (R is then said to be left strongly min-CS) and the dual of each simple right R-module is simple if and only if R is a semiperfect left continuous ring with Soc(RR)=Soc(RR)⊆eRR. Moreover, in this case R is also left Kasch, Soc(eR)≠0 for every local idempotent e of R, and R admits a (Nakayama) permutation of a basic set of primitive idempotents. As a consequence of this result we characterise left PF rings in terms of simple modules over the 2×2 matrix ring by showing that R is left PF if and only if M2(R) is a left strongly min-CS ring such that the dual of every simple right module is simple.

We present an algorithm for computing the Green function of the weighted biharmonic operator Δ|P′|−2Δ on the unit disc (with Dirichlet boundary conditions) for rational functions P. As an application, we show that if P is a Blaschke product with two zeros α1, α2 the Green function is positive if and only if |(α1−α2)/(1−{\bar α}1α2)|≤{2 \over 7}{\sqrt 10}, and also obtain an explicit formula for the Green function of the operator Δ|G|−2Δ, where G is the canonical zero-divisor of a finite zero set on the Bergman space.

We prove that the conditions R·R=0 and R·S=0 are equivalent for hypersurfaces of a 5-dimensional semi-Riemannian space form N5(c). This solves a problem by P.J. Ryan in the case of hypersurfaces of dimension 4 in semi-Riemannian space forms.

A subgroup M of an infinite group G is said to be nearly maximal if it is a maximal element of the set of all subgroups of G having infinite index; i.e. if the index |G:M| is infinite but every subgroup of G properly containing M has finite index in G. The near Frattini subgroup ψ(G) of an infinite group G can now be defined as the intersection of all nearly maximal subgroups ofG, with the stipulation that ψ(G)=G if G has no nearly maximal subgroups. These concepts have been introduced by Riles [5]. It was later proved by Lennox and Robinson [4] that a finitely generated soluble-by-finite group G is infinite-by-nilpotent if and only if all its nearly maximal subgroups are normal. It follows that in the class of finitely generated soluble-by-finite groups the property of being finite-by-nilpotent is inherited from the near Frattini factor group G/ψ(G) to the group G itself. In the study of ordinary Frattini properties of infinite groups, some analogies exist between the behaviour of finitely generated soluble groups and soluble minimax residually finite groups (see for instance [6] and [7]). This fact could suggest that a result corresponding to that of Lennox and Robinson also holds for soluble residually finite minimax groups. Unfortunately in this case the property of being finite-by-nilpotent cannot be detected from the behaviour of nearly maximal subgroups, this phenomenon depending on the fact that infinite soluble residually finite minimax groups may be poor of such subgroups.

If A is an algebra and ϑ is a congruence on A then A is said to be ϑ-coherent provided that, for every subalgebra B of A, if B contains some ϑ-class then B is a union of ϑ-classes. An algebra A is said to be congruence coherent if it is ϑ-coherent for every ϑ∈>ConA. This notion was investigated by Beazer [2] in the context of de Morgan algebras. Specifically, he showed that a de Morgan algebra is congruence coherent if and only if it is boolean, or simple, or the 4-element de Morgan chain. He also showed that if an algebra in the Berman class K1,1 of Ockham algebras is congruence coherent then it is necessarily a de Morgan algebra; and that a p-algebra is congruence coherent if and only if it is boolean. This notion has also been considered in the context of distributive double p-algebras by Adams, Atallah and Beazer [1] who showed that particular examples of congruence coherent double p-algebras are those that are congruence regular (in the sense that if two congruences have a class in common then they coincide). In this paperNATO Collaborative Research Grant 960153 is gratefully acknowledged. we extend the results of Beazer to the class of double MS-algebras.