All groups considered are finite. In recent years a number of generalizations of the classic Jordan-Hölder Theorem have been obtained (see [7], Theorem A.9.13): in a finite group G a one-to-one correspondence as in the Jordan-Holder Theorem can be defined preserving not only G-isomorphic chief factors but even their property of being Frattini or non-Frattini chief factors. In [2] and [13] a new direction of generalization is presented: the above correspondence can be defined in such a way that the corresponding non-Frattini chief factors have the same complement (supplement).

Much interest has been shown in determining the range of values of c for which the sequence [n]c contains infinitely many primes. The result is an elementary deduction from the prime number theorem, of course, of 0<c≤l. In 1953, Piatetski–Shapiro [9] showed that

for 1<c<12/11, where xc(X) stands for the number of primes in the set {[nc]n≤x}.

The Dedekind η-function is defined by

where τ lies in the upper half plane ℋ = {tau;|Im(τ) > 0}, and x = e2πiτ. It is a modular form of weight ½ with a multiplier system. We define an η-product to be a function f (τ) of the form

where rδ ε ℤ. This is a modular form of weight with a multiplier system. The Fourier coefficients of η-products are related to many well-known number-theoretic functions, including partition functions and quadratic form representation numbers. They also arise from representations of the “monster” group [3] and the Mathieu group M24 [13]. The multiplicative structure of these Fourier coefficients has been extensively studied. Recent papers include [1], [4], [5] and [6]. Here we study the connections between the density of the non-zero Fourier coefficients of f(τ) and the representability of f(τ) as a linear combination of Hecke character forms (defined in Section 4 below). We first make the following definition.

The Lindenstrauss formula

which states a strong relationship between the (Clarkson) modulus of uniform convexity δx of a Banach space X and the modulus of uniform smoothness px* of the conjugate space X*, is well known. Following the idea of the definitions of nearly uniform smooth space by S. Prus and modulus of uniform smoothness we define a modulus of nearly uniform smoothness and prove some Lindenstrauss type formulae concerning this modulus and the modulus of nearly uniform convexity for some measures of noncompactness.

Over fields of characteristic zero, there are well known constructions of the irreducible representations, due to A. Young, and of irreducible modules, called Specht modules, due to W. Specht, for the symmetric groups Sn which are based on elegant combinatorial concepts connected with Young tableaux etc. (see, e.g. [13]). James [12] extended these ideas to construct irreducible representations and modules over an arbitrary field. Al-Aamily, Morris and Peel [1] showed how this construction could be extended to cover the Weyl groups of type Bn. In [14] Morris described a possible extension of James' work for Weyl groups in general. Later, the present author and Morris [8] gave an alternative generalisation of James' work which is an extended improvement and extension of the original approach suggested by Morris. We now give a possible extension of James' work for finite reflection groups in general.

The aim of this paper is to generalise the results of [7] from the prime to the semiprime case. It was shown, for instance, that if M is the annihilator of a simple right module S of projective dimension 1 over a Noetherian prime polynomial identity (PI) ring R then M is either an invertible ideal or an idempotent ideal [7, Proposition 4.2]. One of the main applications of this result was that a prime Noetherian affine PI ring of global dimension less than or equal to 2 is a finite module over its centre. It turns out that this theorem is valid more generally when the ring is semiprime [1, Theorem A]. Clearly this requires [7, Proposition 4.2] also to be strengthened to the semiprime case. We do this by showing that a right invertible maximal ideal in a semiprime Noetherian PI ring is also left invertible (Theorem 3.5).

All rings in this paper are associative but not necessarily with an identity. The ring R with an identity adjoined will be denoted by R#.

To denote that I is an ideal (right ideal, left ideal) of a ring R we write I ◃ R (I <rR, I <1R).

A ring R is called right (left) fully idempotent if for every I <rR (I<1R), I = I2.

At the conference “Methoden der Modul und Ringtheorie” in Oberwolfach, Germany in 1993, J. Clark raised the question as to whether every right fully idempotent ring is left fully idempotent (see also [3]). A similar question was raised by S. S. Page in [5]. In this note we answer the questions in the negative.

We start with some general observations most of which are perhaps well known. We include their simple proofs for completeness.

Bass [1] proved that if R is a left perfect ring, then R contains no infinite sets of orthogonal idempotents and every nonzero left R-module has a maximal submodule, and asked if this property characterizes left perfect rings ([1], Remark (ii), p. 470). The fact that this is true for commutative rings was proved by Hamsher [12], and that this is not true in general was demonstrated by examples of Cozzens [7] and Koifman [14]. Hamsher's result for commutative rings has been extended to some noncommutative rings. Call a ring left duo if every left ideal is two-sided; Chandran [5] proved that Bass’ conjecture is true for left duo rings. Call a ring R weakly left duo if for every r ε R, there exists a natural number n(r) (depending on r) such that the principal left ideal Rrn(r) is two-sided. Recently, Xue [21] proved that Bass’ conjecture is still true for weakly left duo rings.

It is well known that for finite dimensional algebras, “bounded representation type” implies “finite representation type”; this is the assertion of the First Brauer-Thrall Conjecture (hereafter referred to as Brauer-Thrall I), proved by Roiter [26] (see also [23]). More precisely, it states that if R is a finite dimensional algebra over a field k, such that there is a finite upper bound on the k-dimensions of the finite dimensional indecomposable right R-modules, then up to isomorphism R has only finitely many (finite dimensional) indecomposable right modules. The hypothesis and conclusion are of course left-right symmetric in this situation, because of the duality between finite dimensional left and right R-modules, given by Homk(−, k). Furthermore, it follows from finite representation type that all indecomposable R modules are finite dimensional [25].

W. A. Bogley and M. A. Gutierrez [2] have recently obtained an eight-term exact homology sequence that relates the integral homology of a quotient group Г/MN, where M and N are normal subgroups of the group Г, to the integral homology of the free product Г/M * Г/N in dimensions ≤3 by means of connecting terms constructed from commutator subgroups of Г, M, N and M ∩ N. In this paper we use the methods of [4] to recover this exact sequence under weaker hypotheses and for coefficients in /q for any non-negative integer q. Further, for q = 0 we extend the sequence by three terms in order to capture the relation between the fourth homology groups.

In this paper we study commutants of Toeplitz operators with polynomial symbols acting on Bergman spaces of various domains. For a positive integer n, let V denote the Lebesgue volume measure on ℂn. If ω is a domain in ℂn, then the Bergman space is defined to be the set of all analytic functions from ω into ℂ such that

Brian Hartley asked me whether a free (nilpotent of class 2 and exponent p2)-group of countable rank has a faithful linear representation of finite degree, p here being a prime of course. The answer is yes. The point is that this then yields via work of F. Leinen and M. J. Tomkinson, see [3,3.6] an image of a linear p-group, which is not even finitary linear. The question of which relatively free groups have faithful linear representations dates back at least to work of W. Magnus in the 1930's, see [4, pp. 33, 34 and the final comment on p. 40] for a discussion of this. Our construction, which works more generally, is a further contribution. We write ℜc for the variety of nilpotent groups of class at most c and (ℭ9 for the variety of groups of exponent dividing q.

A kernel functor (equivalently, a left exact torsion preradical) is a left exact subfunctor of the identity on the category R-mod of left R-modules over a ring R with identity. A kernel functor is said to be idempotent if, in addition, σ satisfies σ(M / σ(M)) = 0 for every M ∊ R-mod. To every kernel functor / there corresponds a unique topologizing filter ℒσ = {I Ⅰ σ (R/I) = R/I} of left ideals and a unique class ℱσ = {M ∊ R-mod Ⅰ σ(M) = M} that is closed under homomorphic images, submodules, and direct sums. The idempotence of σ is characterized by either of the following additional conditions:

(1) if I ∊ ℒσ, K ⊆ I, and (K:x) = {r ∊ R ∣ rx ∊ K} ∊ ℒσ for each x ∊ I, then K ∊ ℒ or

(2) ℱσ is closed under extensions of one member of ℱσ by another member of ℱσ Idempotent kernel functors are important since they are the tool used to construct localization functors. For M∊ R-mod, let E(M) denote the injective hull of M. A kernel functor σ is called stable if Mℱ implies that E(M) ∊ ℱσ For more information about kernel functors, see [6], [7], [14], and [15].

M. E. Adams and Matthew Gould [1] have obtained a remarkable classification of ordered sets P for which the monoid End P of endomorphisms (i.e. isotone maps) is regular, in the sense that for every f є End P there exists g є End P such that fgf = f. They show that the class of such ordered sets consists precisely of

(a) all antichains;

(b) all quasi-complete chains;

(c) all complete bipartite ordered sets (i.e. given non-zero cardinals α β an ordered set Kα,β of height 1 having α minimal elements and β maximal elements, every minimal element being less than every maximal element);

(d) for a non-zero cardinal α the lattice Mα consisting of a smallest element 0, a biggest element 1, and α atoms;

(e) for non-zero cardinals α, β the ordered set Nα,β of height 1 having α minimal elements and β maximal elements in which there is a unique minimal element α0 below all maximal elements and a unique maximal element β0 above all minimal elements (and no further ordering);

(f) the six-element crown C6 with Hasse diagram

A similar characterisation, which coincides with the above for sets of height at most 2 but differs for chains, was obtained by A. Ya. Aizenshtat [2].

Let k be a field, let R be a noetherian k-algebra of finite Gelfand-Kirillov dimension GK(R), and let M be a finitely generated right R-module. A standard prime factor series for M is a finite sequence of submodules 0 = N0 ⊂ N1 ⊂…⊂ Ni−1 ⊂ Ni ⊂.… ⊂ Nn = M, such that for each i the annihilator Pi = rR (Ni/Ni−1) is the unique associated prime of Ni/Ni−1 and GK(R/Pi)≤ GK(R/Pj) whenever i≤ j. The set of prime ideals arising from such a series is an invariant of M, called the set of standard primes St(M) of M. The concept, inspired by the notion of a standard affiliated series introduced by Lenagan and Warfield in [7], has been developed in [5], where it was shown that St(M) coincides with the set of all those prime ideals that are minimal over the annihilator of a nonzero submodule of M.

Let Fg be a closed orientable surface of genus g > 1 and let be the Teichmuller space of Fg, i.e., the space of marked hyperbolic structures on Fg We shall also denote by the space of marked hyperbolic structures on Fgwith one distinguished point; by this, we mean a distinguished point on the universal cover gof Fg. This space is isomorphic to the space of marked complete hyperbolic structures on a genus g surface with 1 cusp which is the usual interpretation of . Choose a decomposition of Fginto pairs of pants by a collection of non–intersecting, totally geodesic simple closed curves. The Fenchel–Nielsen coordinates for relative to this decomposition are given by the lengths of the curves as well as twist parameters defined on each curve. Varying the length and twist parameters gives deformations of the marked hyperbolic structures.

In [3], a group G was said to be a CF-group if, for every subgroup H of G, H/CoreGH is finite. It was shown there that a locally finite CF-group G is abelian-by-finite and that there is a bound for the indices |H: CoreGH| as H runs through all subgroups of G. (Groups for which such a bound exists were referred to in [3] as BCF-groups.) The CF-property was further investigated in [10], one of the main results there being that nilpotent CF-groups are (again) abelian-by-finite and BCF. In the present paper, we shall discuss the CF-property in conjunction with some related properties, which are defined as follows.

The concept of a Hilbert module (over an H*-algebra) arises as a generalization of that of a complex Hilbert space when the complex field is replaced by an (associative) H*-algebra with zero annihilator. P. P. Saworotnow [13] introduced Hilbert modules and extended to its context some classical theorems from the theory of Hilbert spaces, J. F. Smith [17] gave a complete structure theory for Hilbert modules, and G. R. Giellis [9] obtained a nice characteristization of Hilbert modules.

Let R be a commutative ring (with non-zero identity) and let M be an R-module.

In this paper we refer to [13] and [16] for the basic terminology and properties of Noetherian rings. For example, an FBNring means a fully bounded Noetherian ring [13, p. 132], and a cliqueof a Noetherian ring Rmeans a connected component of the graph of links of R[13, p. 178]. For a ring Rand a right or left R–module Mwe use pr.dim.(M) and inj.dim.(M) to denote its projective dimension and injective dimension respectively. The right global dimension of Ris denoted by r.gl.dim.(R).