In this paper, several characterizations are given of the group of eigenvalues of a rank one transformation. One of these is intimately related to the corresponding expression for the maximal spectral type of a rank one transformation given in an earlier paper.

We consider operators T of the form Tf = {Tjfj}, where Tjfj(x) = (p. v) ∫Rn kj(x — y)fj(y) dy. Under appropriate conditions on the kj, two-weighted estimates for T are obtained, the weights being radial and suitably linked.

Necessary and sufficient conditions are given for a generalised matrix ring to be semiprime right Goldie.

Quasi-hereditary algebras were introduced by Cline-Parshall-Scott (see [CPS] or [PS]) to deal with highest weight categories which occur in the study of semi-simple complex Lie algebras and algebraic groups. In fact, the quasi-hereditary algebras which appear in these applications enjoy a number of additional properties. The objective of this brief note is to describe a class of lean quasi-hereditary algebras [ADL] which possess such typical characteristics. A study of these questions originated in collaboration with C. M. Ringel (see [DR]).

Nous proposons de construire des polynômes à coefficients p-adiques, de degré D, sans terme constant, ayant plus que 2D + 1 variables. Ces polynômes n'ont dans Zp que le zéro trivial modulo p2. Le but que nous cherchons à atteindre, à partir des polynômes ainsi construit, est de minorer les valeurs de p pour lesquelles l'équation F(x1,...,X2D+1) ≡ 0 (mod p2) admet une solution primitive, F est un élément de Zp[X1,... ,X2D+1] de degré D sans term constant.

We prove that a free centred extension R[E] is a semisimple ring if R is a semisimple ring and C[E] is semisimple for every field C which is the extended centroid of a primitive factor of R.

We give a complete description of the second Fox subgroup G ∩ (1 + Δ2(G)Δ(H)) relative to a given normal subgroup H of an arbitrary finitely generated group G.

It is proved that if r* is the weak normalization of an integral domain r, then the weak normalization of the power series ring r[[x1,....xn]] is contained in R*[[X1,....Xn]]. Consequently, if R is a weakly normal integral domain, then R[[X1,....Xn]] is also weakly normal.

For the regular representation of a pseudoreflection group G we characterize the occurrences of the contragredient representation as the gradient spaces of a set of Chevalley generators of the invariants of G.

We give a characterisation of where and are subspace lattices with commutative and either completely distributive or complemented. We use it to show that Lat is a CSL algebra with a completely distributive or complemented lattice and is any operator algebra.

Let S = k[[x,y]] be a formal power series ring in two variables x, y over a field k and I an (x, y)-primary ideal of S. We show that S/I is selfinjective if Hi(S/I, S/I ⊗k S/I) = 0 for i = 1 and 2.

Let RG denote the group ring of a group G over a semiprime ring R. We prove that, if the center of G is of finite index and some natural restrictions hold, then every R-derivation of RG is inner. We also give an example of a group G which is both locally finite and nilpotent and such that, for every field F, there exists an F-derivation of FG which is not inner.

Let D be an integral domain, and let X be an analytic indeterminate. As usual, if I is an ideal of D, set It = ∪{JV = (J-1)-1 | J is a nonzero finitely generated subideal of I}; this defines the t-operation, a particularly useful star-operation on D. We discuss the t-operation on R[[X]], paying particular attention to the relation between t- dim(R) and t- dim(R[[X]]). We show that if P is a t-prime of R, then P[[X]] contains a t-prime which contracts to P in R, and we note that this does not quite suffice to show that t- dim(R[[X]]) ≥ t- dim(R) in general. If R is Noetherian, it is easy to see that t- dim(R[[X]]) ≥ t- dim(R), and we show that we have equality in the case of t-dimension 1. We also observe that if V is a valuation domain, then t-dim(V[[X]]) ≥ t- dim(V), and we give examples to show that the inequality can be strict. Finally, we prove that if V is a finite-dimensional valuation domain with maximal ideal M, then MV[[X]] is a maximal t-ideal of V[[X]].

Special cases of Bovdi's conjecture are proved. In particular the conjecture is proved for supersolvable and Frobenius groups. We also prove that if is finite, α ∊ VℤG a torsion unit and m the smallest positive integer such that αm ∊ G then m divides .

Cet article de survol est le résumé de la conférence Coxeter-James de l'auteur, prononcée à la réunion d'hiver 1993 de la Société Mathématique du Canada.

La théorie de Morse décrit les liens entre la topologie d'une variété et la topologie des points critiques d'une fonction sur cette variété. La fonctionnelle d'énergie pour les applications d'une surface dans une variété, dont les points critiques seront des applications harmoniques et parfois holomorphes, et la fonctionnelle de Yang-Mills pour des connections sur une variété de dimension quatre sont deux cas en dimension infinie pour lesquels la théorie de Morse ne tient pas. Néanmois, dans les deux cas, on peut récupérer une quantité étonnante d'information, pourvu qu'on stabilise par rapport à un degré ou une charge qui sont des données du problème. Les preuves recyclent des résultats de la théorie de l'homotopie des années '70, et les combinent à des idées de géométrie complexe pour donner de jolis modèles des espaces en cause en termes de "particules". Nous espérons donner un survol général et accessible des idées utilisées.

L'objet de cet article est d'une part la comparaison de la mesure de Mahler et de la longueur d'un polynôme à coefficients entiers dont toutes les racines sont réelles positives. Nous comparons ensuite la mesure de Mahler d'un polynôme à coefïîcients entiers ayant toutes ses racines réelles à une mesure généralisant la longueur.

A description is given of the unit group for the two groups G = D12 and G = D8 × C2. In particular, it is shown that in both cases the bicyclic units generate a torsion-free normal complement. It follows that the Bass-cyclic units together with the bicyclic units generate a subgroup of finite index in for all n ≥ 3.

In this paper we obtain the asymptotic expansions of the Carathéodory and Kobayashi metrics of strictly pseudoconvex domains with C∞ smooth boundaries in ℂn. The main result of this paper can be stated as following:

Main Theorem. Let Ω be a strictly pseudoconvex domain with C∞ smooth boundary. Let FΩ(z,X) be either the Carathéodory or the Kobayashi metric of Ω. Let δ(z) be the signed distance from z to ∂Ω with δ(z) < 0 for z ∊ Ω and δ(z) ≥ 0 for z ∉ Ω. Then there exist a neighborhood U of ∂Ω, a constant C > 0, and a continuous function C(z,X):(U ∩ Ω) × ℂn -> ℝ such that

and|C(z,X)| ≤ C|X| for z ∊ U ∩ Ω and X ∊ ℂn

Let R be a semiprime ring with a derivation d, λ a left ideal of R and k, n two positive integers. Suppose that [d(xn),xn]k = 0 for all x ∊ λ. Then [λ,R]d(R) = 0. That is, there exists a central idempotent e ∊ U, the left Utumi quotient ring of R, such that d vanishes identically on eU and λ(l — e) is central in (1 — e)U

In this paper, we show that the lattice of closed subsets of the completion, in the Jacobson radical topology, of a finitely generated module M is isomorphic to the completion, under the Hausdorff topology, of the lattice of closed subsets of M. This extends submodule-theoretic results for complete modules to modules satisfying Chevalley's Theorem. We show that the lattice of submodules of every finitely generated module over a semi-local ring R is complete in the Hausdorff topology if and only if R is complete in the Jacobson radical topology.