In the study of certain prime Noetherian rings it is natural to consider the set C of elements which are regular modulo all height-1 prime ideals of R. For R commutative, this set C is simply the set of units. In general this is not the case, though with certain additional conditions we can state non-commutative versions of the Principal Ideal Theorem.

The purpose of this note is to present certain aspects of the theory of spectral operators in Grothendieck spaces with the Dunford-Pettis property, briefly, GDP-spaces, thereby elaborating on the recent note [10].

For example, the sum and product of commuting spectral operators in such spaces are again spectral operators (cf. Proposition 2.1) and a continuous linear operator is spectral if and only if it has finite spectrum (cf. Proposition 2.2). Accordingly, if a spectral operator is of finite type, then its spectrum consists entirely of eigenvalues. Furthermore, it turns out that there are no unbounded spectral operators in such spaces (cf. Proposition 2.4). As a simple application of these results we are able to determine which multiplication operators in certain function spaces are spectral operators.

Let X be a complex Banach space. We denote by B(X) the algebra of all bounded linear operators on X. Let = (T1, …, Tn) be a commuting n-tuple of operators on X. And let στ() and σ″() by Taylor's joint spectrum and the doubly commutant spectrum of , respectively. We refer the reader to Taylor [8] for the definition of στ() and σ″(), A point z = (z1,…, zn) of ℂn is in the joint approximate point spectrum σπ() of if there exists a sequence {xk} of unit vectors in X such that

∥(Ti – zi)xk∥→0 as k → ∞ for i = 1, 2,…, n.

V-rings and their generalisations have been studied extensively in recent years [2], [3], [5],[6], [7]. All the rings we consider will be associative rings with 1 ≠ 0 and all the modules considered will be unitary left R-modules. All the concepts will be left-sided unless otherwise mentioned. Thus by an ideal in R we mean a left ideal of R. A ring R is called a V-ring (respectively a GV-ring) if every simple (resp. simple, singular) module is injective. An R-module M is called p-injective if any homomorphism f: I → M with I a principal left ideal of R can be extended to a homomorphism g: R → M. A ring R is called a p-V-ring (resp. a p-V'-ring) if every simple (resp. simple, singular) module over R is p-injective. The object of the present paper is to introduce torsion theoretic generalizations of p-V-rings and prove results similar to those obtained by Yue Chi Ming about p-V-rings and p-V'-rings [6], [7]. For any M ∈ R-mod, J(M) will denote the Jacobson radical of M and Z(M) the singular submodule of M. For any λ ∈ R, we denote the left annihilator { r ∈ R| rλ =0 } of λ in R by l(λ).

Let A and B be semisimple Banach algebras, and let M1(A) (resp. M1(B)) be the algebra of left multipliers on A (resp. B). Suppose that A is an abstract Segal algebra in B. We find conditions on A and B which imply that M1(A) is topologically algebra isomorphic to M1(B). As a special case we obtain the result of [8] which states that if A is an A*-algebra that is a*-ideal in its B*-algebra completion B and A2 is dense in A then M1(A) is topologically algebra isomorphic to M1(B). We make an application of our main result to right complemented Banach algebras.

In a recent paper [3], D. W. Lewis proposed the following conjecture. (The notation is the same as that in [2] and [3].)

Conjecture. Let F be a field of characteristic not 2 and let a1, b1…, an, bn ∈ Fx. The tensor product of quaternion algebras

is a division algebra if and only if the quadratic form over F

is anisotropic.

This equivalence indeed holds for n = 1 as is well known [2, Theorem 2.7], and Albert [1] (see also [4, §15.7]) has shown that it also holds for n = 2. The aim of this note is to provide counterexamples to both of the implications for n ≥ 3.

Let M be a differentiable manifold of dimension m. A tensor field f of type (1, 1) on M is called a polynomial structure on M if it satisfies the equation:

where a1, a2, …, an are real numbers and I denotes the identity tensor of type (1, 1).

We shall suppose that for any x ∈ M

is the minimal polynomial of the endomorphism fx: TxM → TxM.

We shall call the triple (M, f, g) a polynomial Lorentz structure if f is a polynomial structure on M, g is a symmetric and nondegenerate tensor field of type (0, 2) of signature

such that g (fX, fY) = g(X, Y) for any vector fields X, Y tangent to M. The tensor field g is a (generalized) Lorentz metric.

This paper is a sequel to [2]. A polycyclic-by-finite group G was there called dihedral free if G contains no subgroup isomorphic to 〈b, a:ba = b-1 a2 = 1〉 whose normalizer has finite index in G. It was shown in [2, Theorem F] that, if R is a commutative Noetherian domain, the group ring RG is a prime Noetherian maximal order if and only if R is integrally closed, G is dihedral free, and G has no non-trivial finite normal subgroups. Throughout, R and G will be assumed to satisfy these hypotheses. The main aim of the paper is to study the class group of the maximal order RG.

Throughout this paper all rings are assumed commutative with identity. Among integral domains, Dedekind domains are characterized by the property that every ideal is a product of prime ideals. For a history and proof of this result the reader is referred to Cohen [2, pp. 31–32]. More generally, Mori [5] has shown that a ring has the property that every ideal is a product of prime ideals if and only if it is a finite direct product of Dedekind domains and special principal ideal rings (SPIRS). Rings with this property are called general Z.P.I.-rings.

The trivial observation that every automorphism of a group is determined by its restriction to a set of generators suggests the converse question: if X is a subset of a group G such that each automorphism of G is determined (or “almost” determined) by its restriction to X, to what extent is the structure of G governed by that of the subgroup which X generates? Is this subgroup in some sense necessarily “large” in G? If the index of the subgroup is used as a measure of largeness, then in the absence of additional hypotheses, the answer to the second question is generally “no”, the additive group of rationals with X = {1} being an obvious counterexample. (More confounding is the existence of uncountable torsion-free abelian groups for which inversion is the only non-trivial automorphism. See, for example, [2], [3], and [4].) However, under certain finiteness assumptions, it seems that some positive conclusions are obtainable. One such example will be considered here.

On p. 244 of Glasgow Math. J.27 (1985) on the right hand side of one of the 6 equations characterizing the 4 fixed points of the involution v a sign error has occurred.The relevant equation should read

y0y3y5y6=–1,

or the points would not lie on the curve.

Correcting the error unfortunately invalidates the model of an elliptic curve given in §6, which therefore has to be re-evaluated. First we find, in the notation of the paper,

2 f (x) = ((r + 1)/(R + 2))2.

In the theory of spectral (and prespectral) operators in a Banach space or in a locally convex topological vector space the countable additivity (in some topology) of a resolution of the identity of the operator is a standing assumption. One might wonder why. Even if one cannot completely agree with the opinion of Diestel and Uhl ([6, p. 32]) stating that “countable additivity [of a set function] is often more of a hindrance than a help”, it might be interesting to study which portions of the theory of (pre)spectral operators and in which form extend to the more general situation described below.

The purpose of this short note is to make an observation about Dunford–Pettis operators from L1[0, 1] to C0. Recall that an operator T:E→F (where E and F are Banach spaces) is called Dunford–Pettis if T takes weakly convergent sequences of E into norm convergent sequences of F. A Banach space F has the Compact Range Property (CRP) if every operator T:L1]0, 1]→F is Dunford–Pettis. Talagrand shows in his book [2] that C0 does not have the CRP. It is of interest (especially in mathematical economics [3]) to note that every positive operator from L1[0, 1] to C0 is Dunford–Pettis.