The class of $\mathfrak{m}$-full and four related classes of ideals in a local ring (R, $\mathfrak{m}$) are extended by replacing $\mathfrak{m}$ with other ideals and the resulting classes of ideals are compared. It is shown that contracted ideals are $\mathfrak{m}$-full in a local ring with infinite residue field.

In this brief note, we evaluate certain quartic and octic Gauss sums with the use of theorems on fourth and eighth power difference sets. We recall that a subset H of a finite (additive) abelian group G is said to be a difference set of G [5, p. 64] if for some fixed natural number λ, every nonzero element of G can be written as a difference of two elements of H fin exactly λ ways.

It has been proved by D. E. Cohen [1] that the lattice of all varieties of metabelian groups is countable. In this paper, we show that the lattice of all varieties of completely simple semigroups with metabelian subgroups has the cardinality of the continuum. M. Petrich and N. R. Reilly have introduced in [6] the notion of near varieties of idempotent generated completely simple semigroups. The mapping assigning to every variety $\mathcal{V}$ of completely simple semigroups the class of all idempotent generated members of $\mathcal{V}$ is a complete lattice homomorphism of the lattice of all varieties of completely simple semigroups onto the lattice of all near varieties of idempotent generated completely simple semigroups. In this paper we show that, in fact, the lattice of all near varieties of idempotent generated completely simple semigroups with metabelian subgroups has itself the cardinality of the continuum.

The paper is related to the area which was recently called topological homology [3, 6, 12, 16, 4]. We consider questions associated with the central Hochschild cohomology of C*-algebras. The study of the latter was begun by J. Phillips and I. Raeburn in [9, 10], when they were investigating some problems of the theory of perturbations of C*-algebras. In [8] we obtained a description of the structure of C*-algebras with central bidimension zero: it was proved that these C*-algebras are unital and have continuous trace. In the special case of separable and a priori unital C*-algebras this statement was proved by J. Phillips and I. Raeburn in [11] with the help of a different approach. The question was raised. Which values can the central bidimension of C*-algebras take? In the present paper it is shown that, for any CCR-algebra A having at least one infinite-dimensional irreducible representation, the central bidimension and the global central homological dimension of A are greater than one. At the same time it is proved that there exist CCR-algebras which are centrally biprojective, but which have both dimensions equal to one. This situation contrasts with the state of affairs in the “traditional” theory of the Banach Hochschild cohomology. Recall [3, Ch. 5] that the bidimension and the global homological dimension of any infinite-dimensional biprojective C*-algebra are equal to two. Besides, there is no CCR-algebra of bidimension one (respectively, global homological dimension one). See [7].

1. Equation (4) should read

2. On p. 83 below equation (7) replace

3. On p. 83 two lines above equation (9) replace

A ring R is called radical if it coincides with its Jacobson radical, which means that Rforms a group under the operation a ° b = a + b + ab for all a and b in R. This group is called the adjoint group R° of R. The relation between the adjoint group R° and the additive group R+ of a radical rin R is an interesting topic to study. It has been shown in [1] that the finiteness conditions “minimax”, “finite Prufer rank”, “finite abelian subgroup rank” and “finite torsionfree rank” carry over from the adjoint group to the additive group of a radical ring. The converse is true for the minimax condition, while it fails for all the other above finiteness conditions by an example due to Sysak [6] (see also [2, Theorem 6.1.2]). However, we will show that the converse holds if we restrict to the class of nil rings, i.e. the rings R such that for any a є R there exists an n = n(a) with an = 0.

This paper investigates topological reconstruction, related to the reconstruction conjecture in graph theory. We ask whether the homeomorphism types of subspaces of a space X which are obtained by deleting singletons determine X uniquely up to homeomorphism. If the question can be answered affirmatively, such a space is called reconstructible. We prove that in various cases topological properties can be reconstructed. As main result we find that familiar spaces such as the reals ℝ, the rationals ℚ and the irrationals ℙ are reconstructible, as well as spaces occurring as Stone–Čech compactifications. Moreover, some non-reconstructible spaces are discovered, amongst them the Cantor set C.

Let A be a Banach *-algebra. We assume neither that A has an identity element, nor that the involution on A is continuous.

The following two results in the theory of division algebras are well known and easily proved, for an arbitrary commutative field k (cf. for example [3, Chapter 10]).

(i) The tensor product of two central division algebras over k of coprime degrees is again a division algebra.

(ii) Every central division algebra over k is a tensor product of division algebras of prime power degrees.

It is natural to ask whether corresponding results hold for commutative fields. The answers are not hard to find but (as far as I am aware) have not appeared in print before; since they throw some light on the nature of tensor products they seemed worth recording.

We analyze the computational complexity of an algorithm to solve the conjugacy search problem in a certain family of metabelian groups. We prove that in general the time complexity of the conjugacy search problem for these groups is at most exponential. For a subfamily of groups, we prove that the conjugacy search problem is polynomial. We also show that for a different subfamily the conjugacy search problem reduces to the discrete logarithm problem.

In 1959, Bishop [4] published a seminal paper in which he studied various types of spectral decompositions or “duality theories” that an arbitrary bounded linear operator on a reflexive Banach space might have. In the course of his investigations, he isolated the following analytic property which he called condition (β).

We give upper and lower bounds for the number of rational points on Prym varieties over finite fields. Moreover, we determine the exact maximum and minimum number of rational points on Prym varieties of dimension 2.

For a completely 0-simple semigroup, Howie [2] has investigated the subsemigroup generated by the idempotents. Here we determine those elements of such a semigroup which are generated by the set of nilpotent elements and hence we derive a condition for a completely 0-simple semigroup to be nilpotent generated. This condition is purely combinatorial, in terms of the structure of the graph associated with the semigroup, and it includes the case of a non-regular Rees matrix semigroup.

Throughout this paper S will denote a given monoid, that is, a semigroup with an identity. A set A is a right S-system if there is a map φ: A × S → A satisfying

for any element a of A and any elements s, t of S. For φ(a, s) we write as and we refer to right S-systems simply as S-systems. One has the obvious definitions of an S-subsystem and an S-homomorphism.

Let X and Y be Banach spaces such that each one is isomorphic to a complemented subspace of the other. In 1996, W. T. Gowers solved the Schroeder-Bernstein Problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we give suitable conditions on X, Y, the supplemented subspaces of Y in X and of X in Y to yield that X is isomorphic to Y. In other words, we obtain generalizations of Pełczyński's decomposition method via supplemented subspaces. In order to determine all the possible generalizations, we introduce the notion of Mixed Schroeder-Bernstein Quadruples for Banach spaces. Then, we use some Banach spaces constructed by W. T. Gowers and B. Maurey in 1997 to characterize them.

In this paper, we prove a combination theorem for a complex of relatively hyperbolic groups. It is a generalization of Martin’s (Geom. Topology18 (2014), 31–102) work for combination of hyperbolic groups over a finite MK-simplicial complex, where k ≤ 0.

A monoid S satisfies Condition (A) if every locally cyclic left S-act is cyclic. This condition first arose in Isbell's work on left perfect monoids, that is, monoids such that every left S-act has a projective cover. Isbell showed that S is left perfect if and only if every cyclic left S-act has a projective cover and Condition (A) holds. Fountain built on Isbell's work to show that S is left perfect if and only if it satisfies Condition (A) together with the descending chain condition on principal right ideals, MR. We note that a ring is left perfect (with an analogous definition) if and only if it satisfies MR. The appearance of Condition (A) in this context is, therefore, monoid specific. Condition (A) has a number of alternative characterisations, in particular, it is equivalent to the ascending chain condition on cyclic subacts of any left S-act. In spite of this, it remains somewhat esoteric. The first aim of this paper is to investigate the preservation of Condition (A) under basic semigroup-theoretic constructions. Recently, Khosravi, Ershad and Sedaghatjoo have shown that every left S-act has a strongly flat or Condition (P) cover if and only if every cyclic left S-act has such a cover and Condition (A) holds. Here we find a range of classes of S-acts $\mathcal{C}$ such that every left S-act has a cover from $\mathcal{C}$ if and only if every cyclic left S-act does and Condition (A) holds. In doing so we find a further characterisation of Condition (A) purely in terms of the existence of covers of a certain kind. Finally, we make some observations concerning left perfect monoids and investigate a class of monoids close to being left perfect, which we name left$\mathcal{IP}$a-perfect.

The aim of the present paper is the classification of real hypersurfaces M equipped with the condition Al = lA, l = R(., ξ)ξ, restricted in a subspace of the tangent space T p M of M at a point p. This class is large and difficult to classify, therefore a second condition is imposed: (∇ξ l)X = ω(X)ξ + ψ(X)lX, where ω(X), ψ(X) are 1-forms. The last condition is studied for the first time and is much weaker than ∇ξ l = 0 which has been studied so far. The Jacobi Structure Operator satisfying this weaker condition can be called generalized ξ-parallel Jacobi Structure Operator.

We give a method allowing the generalization of the description of trace spaces of certain classes of holomorphic functions on Carleson sequences to finite unions of Carleson sequences. We apply the result to different classes of spaces of holomorphic functions such as Hardy classes and Bergman type spaces.

Let $(X,\mathcal{F},\mu)$ be a complete probability space and let $\mathcal{B}$ be a sub-$\sigma$-algebra of $\mathcal{F}$. We consider the extreme points of the closed unit ball $\mathbb{B}(\mathcal{A})$ of the normed space $\mathcal{A}$ whose points are the elements of $L^\infty(X,\mathcal{F},\mu)$ with the norm$\Vert\, f \Vert = \Vert\Phi(\vert\, f \vert)\Vert_\infty$, where $\Phi$ is the probabilistic conditional expectation operator determined by $\mathcal{B}$. No $\mathcal{B}$- measurable function is an extreme point of the closed unit ball of $\mathcal{A}$, and in certain cases there are no extreme points of $\mathbb{B}(\mathcal{A})$.

For an interesting family of examples, depending on a parameter $n$, we characterize the extreme points of the unit ball and show that every element of the open unit ball is a convex combination of extreme points. Although in these examples every element of the open ball of radius $\frac{1}{n}$ can be shown to be a convex combination of at most $2n$ extreme points by elementary arguments, our proof for the open unit ball requires use of the $\lambda$-function of Aron and Lohman. In the case of the open unit ball, we only obtain estimates for the number of extreme points required in very special cases, e.g. the $\mathcal{B}$-measurable functions, where $2n$ extreme points suffice.