This paper contains an extension of a result obtained by H. Bart, M. A. Kaashoek and D. C. Lay in (2). These authors studied the reduced algebraic multiplicity RM(A; λ0) of a meromorphic operator function at a point λ0 ∈ C. They proved that under certain conditions this quantity has logarithmic behaviour, i.e.,

For more restricted cases such results had been proved by others, notably I. C. Gohberg and E. I. Sigal (see (4) and (5)). Here we shall prove that such a result also holds for a larger class of operator functions than the diagonable functions considered in (2).

W. H. Cornish (2) has investigated congruences on pseudo-complemented distributive lattices and has identified those ideals (resp. filters) that are congruence kernels (resp. cokernels). In this paper we show that many of the principal results concerning congruence kernels and cokernels hold in a semilattice and therefore do not depend on distributivity, nor on the existence of unions.

If a ∈ L(1, T) for every finite T>1, then we say that the infinite integral a(u)du is convergent with sum s if a(u)du = s. It is well known that a necessary and sufficient condition for a(u)du to be convergent (with some finite sum s) is that Cauchy's criterion,

holds. The object of this note is to obtain a similar result for summability (C, α) of a(u) du which reduces to Cauchy's criterion in the case of convergence. The corresti ponding problem for summable series has been treated by A. F. Andersen in (1).

Groups for which the distributively generated near-ring generated by the endomorphisms is in fact a ring are known as E-groups and are discussed in (3). R. Faudree in (1) has given the only published examples of non-abelian E-groups by presenting defining relations for a family of p–groups. However, as shown in (3), Faudree's group does not have the desired property when p = 2.

Let R be an arbitrary near-ring and define the multiplicative centre Z(R) by

In previous papers (2,3,5) we have established additive or multiplicative commutativity for various near-rings R in which selected elements were restricted to lie in Z(R); the near-rings involved were usually distributively-generated (d-g) and were frequently assumed to have a multiplicative identity element as well.

Let M = SL(2, Z) be the classical modular matrix group. One form of the Poincaré series on M is

here z ∈ H={z = x + iy: y >0}, q ≧ 2 and m ≧ 1 are integers, and the summation is over a complete system of matrices (ab: cd) in M with different lower row. The problem of the identical vanishing of the Poincaré series for different values of m and q goes back to Poincaré.

Radicals appear in many algebraic contents. For modules over a ring, they give rise to pre-torsion and torsion theories, Goldman (5), Lambek (14). In the category of groups, Kurosh, Plotkin and others have introduced radicals (6), (13), (21), but unlike the radicals in module theory these radicals are not necessarily functorial, as for example the nil radical and the Hirsch-Plotkin radical (6). The functorial method in module theory has been extended to abelian categories, Dickson (2), to the category of nilpotent groups, Hilton (8), Warfield (25), and to the category of groups, Plotkin (22), and to general categories, Wiegandt (26), Holcombe and Walker (10).

Clay (3), Johnson (5) and Krimmel (6) have each considered the near-rings with identity on dihedral groups. Krimmel actually generalised the class of dihedral groups and investigated the class of finite non-abelian groups with a cyclic normal subgroup of prime index: we shall call this class . Krimmel considered the near-rings with identity that might be denned on members of and he determined the subclass of groups in which support near-rings of this kind. He also managed to calculate the number of non-isomorphic near-rings involved for certain cases. His methods were essentially combinatorial, and his results were expressed in terms of various integers which characterised the individual members of . Certain features of this work led us to investigate the structure of near-rings on members of from a more algebraic point of view and thereby to complete and extend Krimmel's programme. Part of the work in this paper formed the basis of the second author's thesis (7). We should like to thank Dr. J. Krimmel for permission to include some of his results, and Dr. J. Meldrum who detected an error in our original formulation of Theorem 7.1.

In the representation theory of finite groups, the minimal idempotents of the group algebra play a central role. In this case the minimal idempotents determine irreducible modules over the group algebra, which in turn are in direct correspondence with the irreducible matrix representations of the group; see Chapter IV of the book of C. Curtis and I. Reiner (2). Many of the same ideas generalise to the situation where the group is compact. In addition, minimal idempotents are involved in some important parts of the theory of Hubert algebras; see M. Rieffel's paper (20).

Since Brownian motion is point recurrent in R1, recurrent in R2 and transient in Rn, n ≧ 3 (see (7)), it follows that the total time spent in a bounded open set in R1 or R2 is unbounded. With the following ergodic theorems for Brownian motion in R1 and R2 as motivation, we examine the rate of convergence in these theorems. Note that there is no ergodic property in Rn for n ≧ 3 since Brownian motion is not dense there.

In this paper we study near-rings of functions on Ω-groups which are compatible with all congruence relations. Polynomial functions, for instance, are of this type. We employ the structure theory for near-rings to get results for the theory of compatible and polynomial functions (affine completeness, etc.). For notations and results concerning near-rings see e.g. (10). However, we review briefly some terminology from there. (N, +,.) is a near-ring if (N, +) is a group and . is associative and right distributive over +. For instance, M(A): = (AA, +, °) is a near-ring for any group (A, +) (° is composition). If N is a near-ring then N0: = {n ∈ N/n0 = 0}. A group (Γ, +) is an N-group (we write NΓ) if a “product” ny is defined with (n + n‛)γ = nγ + n‛γ and (nn‛)γ = n(n‛γ). Ideals of near-rings and N-groups are kernels of (N-) homomorphisms. If Γ is a vector-space, Maff (Γ) is the near-ring of all affine transformations on Γ. N is 2-primitive on NΓ if NΓ is non-trivial, faithful and without proper N-subgroups. The (2-) radical and (2-) semisimplicity are defined similarly to the ring case.

Let N be an arbitrary near-ring. Each element a ∈ N determines in a natural way a new multiplication on the elements of N which results in a near-ring Na whose additive group coincides with that of N but whose multiplicative semigroup generally differs. Specifically, we define the product x * y of two elements in Na by x * y = x a y where a product in the original near-ring is denoted by juxtaposition. One easily checks that Na is a near-ring with addition identical to that of N. The original near-ring N will be referred to as the base near-ring, Na will be referred to as a laminated near-ring of N and a will be referred to as the laminating element or sometimes more simply as the laminator.

Suppose F is an additively written free group of countably infinite rank with basis T and let E = End(F). If we add endomorphisms pointwise on T and multiply them by map composition, E becomes a near-ring. In her paper “On Varieties of Groups and their Associated Near Rings” Hanna Neumann studied the sub-near-ring of E consisting of the endomorphisms of F of finite support, that is, those endomorphisms taking almost all of the elements of T to zero. She called this near-ring Φω. Now it happens that the ideals of Φω are in one to one correspondence with varieties of groups. Moreover this correspondence is a monoid isomorphism where the ideals of φω are multiplied pointwise. The aim of Neumann's paper was to use this isomorphism to show that any variety can be written uniquely as a finite product of primes, and it was in this near-ring theoretic context that this problem was first raised. She succeeded in showing that the left cancellation law holds for varieties (namely, U(V) = U′(V) implies U = U′) and that any variety can be written as a finite product of primes. The other cancellation law proved intractable. Later, unique prime factorization of varieties was proved by Neumann, Neumann and Neumann, in (7). A concise proof using these same wreath product techniques was also given in H. Neumann's book (6). These proofs, however, bear no relation to the original near-ring theoretic statement of the problem.

Throughout this note, N will denote a (Left) near-ring with two-sided zero. Definitions of basic concepts can be found in (9).

We prove first that a right ideal I in a d.g. near-ring has a right identity if and only if x ∈ xI for each x ∈I. This enables us to study the structure of regular d.g. near-rings with chain conditions on right annihilators. Specifically we will prove that a regular d.g. near-ring with both the maximum and the minimum conditions on right annihilators is a finite direct sum of near-rings which are either rings of matrices over division rings or non-rings of the form MG(Γ) for a suitable type 2 N-module Γ. Finally we consider the case of maximum condition on N-subgroups.

Nearfields and near-rings are related to many other structures and needed for several representation theorems. Therefore it is important to gain knowledge about the structure of near-rings and nearfields and to find construction methods. The first examples of proper nearfields were constructed by L.E. Dickson 1905, they were finite. 30 years later H. Zassenhaus completely determined the finite nearfields his attention having been attracted to them by the study of certain permutation groups. By the axiomatization of Dickson's methods, H. Karzel succeeded in giving new examples of infinite nearfields. In the extensive generalization by J. Timm the so-called “Dicksonian processes” are the most important tool for constructions of nearfields and near-rings. The report (44) by H. Wähling is a summary of the results on nearfields obtained so far.