The notion of projective Banach module was defined by Helemskii in [1]—the paper which properly founded the homological theory of Banach algebras. The same author introduced the definition of the (relatively) flat Banach module in [2]. Recently M. C. White [3] modified both of those definitions, introducing so called C-projective and C-flat Banach modules.

The concept of cleft extensions, or equivalently of crossed products, for a Hopf algebra is a generalization of Galois extensions with normal basis and of crossed products for a group. The study of these subjects was founded independently by Blattner-Cohen-Montgomery [1] and by Doi-Takeuchi [4]. In this paper, we determine the isomorphic classes of cleft extensions for a infinite dimensional non-commutative, non-cocommutative Hopf algebra kq[X, X–l, Y], which is generated by a group-like element X and a (1,X)-primitive element Y. We also consider the quotient algebras of the cleft extensions.

Let h be the second fundamental form of a compact submanifold M of the Cayley projective plane CaP2. We determine all compact totally complex submanifolds of complex dimension n in CaP2 satisfying |h|2 ≤ n.

Let R be a ring, see below for other notation. The functor categories (mod-R, Ab) and ((R-mod)op, Ab) have received considerable attention since the 1960s. The first of these has achieved prominence in the model theory of modules and most particularly in the investigation of the representation theory of Artinian algebras. Both [11, Chapter 12] and [8] contain accounts of the use (mod-R, Ab) may be put to in the model theoretic setting, and Auslander's review, [1], details the application of (mod-R, Ab) to the study of Artinian algebras. The category ((R-mod)op, Ab) has been less fully exploited. Much work, however, has been devoted to the study of the transpose functor between R-mod and mod-R. Warfield's paper, [13], describes this for semiperfect rings, and this duality is an essential component in the construction of almost split sequences over Artinian algebras, see [4]. In comparison, the general case has been neglected. This paper seeks to remedy this situation, giving a concrete description of the resulting equivalence between (mod-R, Ab) and ((R-mod)op, Ab) for an arbitrary ring R.

Let G be a group. A precrossed G-module is a group homomorphism ∂: M → G together with a group action (g, m) ↦gm of G on M, such that ∂(gm) = g(∂m)g−1. The Peiffer commutator < m, m′ > of two elements m, m′ ∊ M is denned as

< m, m′ >= mm′ m−1(∂mm′)−1

If all Peiffer commutators are trivial, the precrossed G-module is said to be a crossed G-module. The subgroup < M, M > generated by all Peiffer commutators is called the Peiffer subgroup of M; it is the second term of a lower Peiffer central series (see below). The following table indicates how these concepts reduce to more standard concepts when restrictions are placed on ∂ and G.

All groups considered in the sequel are finite. Let (ℭ and denote the formations of groups which consist of collections of groups that respectively either split over each normal subgroup (nC-groups) or for which the groups do not possess nontrivial Frattini chief factors [8]. The purpose of this article is to develop and expand a concept that arises naturally with the residuals for these formations, namely each G-chief factor is non-complemented (Frattini). With respect to a solid set X of maximal subgroups, these properties are generalized respectively to so-called X-parafrattini (X-profrattini) normal subgroups for which each type is closed relative to products. The relationships among the unique maximal normal subgroups that result from these products, the solid set of maximal subgroups X, X-prefrattini subgroups, and the residuals of formations are explored. This leads to a well-defined collected of formations, the partially nonsaturated formations, with properties analogous to those which are totally non-saturated. In the development, attention is given to a set of maximal subgroups which is the image of a solid function defined on all groups, a weaker condition than that of a solid set. A result of particular interest answers affirmatively the long-standing conjecture that a non-trivial nC-group G is solvable if and only if each G-chief factor is complemented by a maximal subgroup. This will force a critical re-examination of the classification problem for nC-groups. Since the article continues the investigations on finite groups initiated in [2], a familiarity with that article is assumed. All other notation and terminology is from [6]. If M is a maximal subgroup of a group G and G/C or e G(M) is a monolithic primitive group, i.e. a group with a unique minimal normal subgroup, then M is called a monolithic maximal subgroupof G.

We prove the pointwise inequality 0 ≥ ρ + ρ⊥ – 1 involving the normalized scalar curvature ρ and normal scalar curvature ρ⊥ of a totally real 3-dimensional submanifold of the nearly Kaehler 6-sphere. Further we classify submanifolds realizing the equality in this inequality.

We study the spectral decomposition with respect to the Jacobi operator, J, of spherical immersions and characterize those with a simple decomposition in terms of the Finite Chen-type submanifolds. As a consequence, we give an application to the inverse problem for J.

It is proved that if every cyclic right R-module is torsionless and R is a left CS-ring then R is semiperfect left continuous with soc(RR)essential in RR. As a consequence every right cogenerator, left CS-ring R is shown to be right pseudo-Frobenius and left continuous, and an example is given to show that R need not be left selfinjective. It is also proved that if R is a left CS-ring and every cyclic right R-module embeds in a free module, then R is quasi-Frobenius if and only if J(R) ⊆ Z(RR).

Let P be a prime ideal of a ring R, O(P) = {a ∊ R | aRs = 0, for some s ∊ R/P} | and Ō(P) = {x ∊ R | xn ∊ O(P), for some positive integer n}. Several authors have obtained sheaf representations of rings whose stalks are of the form R/O(P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P= Ō(P). In this paper we derive various conditions which ensure that a prime ideal P = Ō(P). The property that P = Ō(P) is then used to obtain conditions which determine when R/O(P) has a unique minimal prime ideal. Various generalizations of O(P) and Ō(P) are considered. Examples are provided to illustrate and delimit our results.

In this paper we show that the Weyl spectrum of an operator of class W satisfies the spectral mapping theorem for analytic functions and give the equivalent conditions for an operator of the form normal + compact to be polynomially compact.

A group G is the product of its subgroups A and B if G equals the set AB = (ab | a ∊ A, b ∊ B). A subgroup H of G is called prefactorized if it is the product of a subgroup of A and a subgroup of B; thus H is prefactorized if and only if H = (H ∩ A)(H ∩ B). A prefactorized subgroup H of G is factorized if it contains A ∩ B. if H is any subgroup of G = AB, then the intersection X ofall factorized subgroups of G containing H is itself factorized; see for example [2, Lemma 1.1.2]. This subgroup, which is evidently the smallest factorized subgroup of G which contains H, is called the factorizer of H in G = AB.

We were concerned in [12] with groups G that are isomorphic to all of their non-abelian subgroups. In order to exclude groups with all proper subgroups abelian, which are well understood in the finite case [7] and which include Tarski groups in the infinite case, we restricted attention to the class X of groups G that are isomorphic to their nonabelian subgroups and that contain proper subgroups of this type; such groups are easily seen to be 2-generator, and a complete classification was given in [12, Theorem 2] for the case G soluble. In the insoluble case, G/Z(G) is infinite simple [12; Theorem 1], though not much else was said in [12] about such groups. Here we examine a property which represents a natural generalisation of that discussed above. Let us say that a group G belongs to the class W if G is isomorphic to each of its non-nilpotent subgroups and not every proper subgroup of G is nilpotent. Firstly, note that finite groups in which all proper subgroups are nilpotent are (again) well understood [9]. In addition, much is known about infinite groups with all proper subgroups nilpotent (see, in particular, [8] and [13] for further discussion) although, even in the locally nilpotent case, there are still some gaps in our understanding of such groups. We content ourselvesin the present paper with discussing finitely generated W-groups— note that a W-group is certainly finitely generated or locally nilpotent. We shall have a little more to say about the locally nilpotent case below.

The purpose of this note is to answer in the affirmative a long standing open question raised by Singal and Singal — whether every almost continuous function is closure continuous (θ-continuous).

We use the principle of symmetric criticality to connect the Willmore variational problem for surfaces in a warped product space with base a circle, and the free elastica variational problem for curves on its fiber. In addition we obtain a rational oneparameter family of closed helices in the anti De Sitter 3-space which are critical points of the total squared curvature functional. This means they are free elasticae. Also they are spacelike; this allows us to construct a corresponding family of spacelike Willmore tori in a certain kind of spacetime close to the Robertson-Walker spaces.

On the setting of the unit ball of the complex n-space, some characterizations of M-harmonic Bloch functions are obtained. As an application, Carleson measures are characterized by means of Berezin type integrals of M-harmonic Bloch functions. As one may expect, these results carry over to M-harmonic little Bloch functions and vanishing Carleson measures.

We consider a real-valued function r = M(t) on the real axis, such that M(t) < 0 for t < 0. Under appropriate assumptions on M, the pull-back operator M* gives rise to a transform of Sobolev spaces Ws.p (-∞, 0) that restricts to a transform of Ws.p(-∞, ∞). We construct a bounded linear extension operator Ws.p(-∞, 0) → Ws.p(−∞, ∞), commuting with this transform.

We gave an elementary proof of Theorem 4 in the paper, published in the Glasgow Mathematical Journal 39(1997), 275–284. The result provides an algorithm for approximating the maximal eigenvalue of a nonnegative matrix. Recently the author has learnt that the result can be proved immediately from Theorem 6.8 in [1]. Indeed, the paper [1] determines necessary and sufficient conditions for the convergence of an iterative sequence to the maximal eigenvalue. Their proof needs knowledge of graph theoretical concepts.