In this note we consider the question: If R is a right Noetherian ring and I is an invertible ideal of R, how do the Krull dimensions of various modules, factor rings and over-rings of R, connected with I, compare with the Krull dimension of R? This question is prompted by results in (5) and (6). In comparing the Krull dimension of the ring R with that of the ring R/I, the best result would be that the Krull dimension of the ring R is exactly one greater than that of the ring R/I. This result is not true in general; however, we see, in Theorem 2.4, that if the invertible ideal is contained in the Jacobson radical the result holds. In the general case we find it is necessary to introduce an over-ring T of R generated by the inverse I−1 of I. We then see that the Krull dimension of R is the larger of two possibilities: (a) Krull dimension of R/I plus one or (b) Krull dimension of T. In order to prove this result we construct a strictly increasing map from the poset of right ideals of R to the cartesian product of the poset of right ideals of T with a poset of certain infinite sequences of right ideals of R/I.

A semigroup S with identity is (left) perfect if every unitary left S-system has a projective cover. This is the semigroup analogue of the definition of left perfect rings introduced in (1). The investigation of perfect semigroups was initiated by Isbell (4), who proved that a semigroup is perfect if and only if it satisfies two conditions referred to as conditions A and D.

This note extends to categories Fountain's theorem (2) that for a perfect monoid S, every flat S-set is projective. (The converse is known (4).)

It is a well-known fact that any normed algebra can be represented isometrically as an algebra of operators with the operator norm. As might be expected from the very universality of this property, it is little used in the study of the structure of an algebra. Far more helpful are representations on Hilbert space, though these are correspondingly hard to come by: isometric representations on Hilbert space are not to be expected in general, and even continuous nontrivial representations may fail to exist. The purpose of this paper is to examine a class of representations intermediate in both availability and utility to those already mentioned—namely, representations on reflexive spaces. There certainly are normed algebras which admit isometric representations of the latter type but have not even faithful representations on Hilbert space: the most natural example is the algebra of all continuous linear operators on E where E = lp with 1 < p ≠ 2 < ∞, for Berkson and Porta proved in (2) that if E, F are taken from the spaces lp with 1 < p < ∞ and E ≠ F then the only continuous homomorphism from into is the zero mapping. On the other hand there are also algebras which have no continuous nontrivial representation on any reflexive space—for example the algebra of finite-rank operators on an irreflexive Banach space (see Berkson and Porta (2) or Barnes (1) or Theorem 3, Corollary 1 below).

The well-known algebraic concept of tensor product exists for any variety of algebras.The tensor product of groups and of rings have been studied extensively. For other varieties, such as the variety of semigroups, the tensor product has been investigated more recently (5). In this paper we investigate the tensor product of distributive lattices.

Trevor Evans in (8) introduced postulates for a non-associative number theory similar to, but less general than, those of A. Robinson (9). Evans' number theory is also non-commutative under addition and multiplication, but an alternative equality axiom also suggested by Robinson leads to a number theory which is commutative under addition and still non-associative except in the special case:

We follow the notations and basic equations of Chen (2). Let M be a surface immersed in an m-dimensional space form Rm(c) of curvature c = 1, 0 or −1. We choose a local field of orthonormal frames e1, …, em in Rm(c) such that, restricted to M, the vectors e1, e2 are tangent to M. Let ω1, …, ωm be the field of dual frames. Then the structure equations of Rm(c) are given by

There have been several recent accounts of a theory dual to the well-known theory of primary decomposition for modules over a (non-trivial) commutative ring A with identity: see (4), (2) and (9). Here we shall follow Macdonald's terminology from (4) and refer to this dual theory as “ secondary representation theory ”. A secondary representation for an A-module M is an expression for M as a finite sum of secondary submodules; just as the zero submodule of a Noetherian A-module X has a primary decomposition in X, it turns out, as one would expect, that every Artinian A-module has a secondary representation.

In what follows the term C*-algebra will mean a complex C*-algebra with identity. We denote the identity element by 1. We shall also use the notation and terminology of Dixmier (3) without comment.

A continuous function φ on the unit circle is called badly approximable if ‖ φ − p ‖∞ ≧ ‖ φ |∞ for all polynomials p, where ‖ |∞ is the essential supremum norm. In (4), Poreda asked whether every continuous φ may be written φ = φW+φB, where φW is the uniform limit of polynomials (i.e. φW belongs to the disc algebra A) and φB is badly approximable. We call such a function φ decomposable. In (4), he characterised the badly approximable functions as those of constant non-zero modulus and negative winding number around the origin, i.e. ind (φ)<0. (See (3) for two new proofs of this result.) We show that the answer to Poreda's question is no in general, but give a necessary and sufficient condition for a given φ to have such a decomposition. Then we apply this criterion to solve an interpolation problem.

In their recently published book Bonsall and Duncan ask the following question ((1), p. 65): if A is a Banach star algebra with an identity, is it true that

A bounded linear operator T on a complex reflexive Banach space is said to be well-bounded if it is possible to choose a compact interval J = [a, b] and a positive constant M such that

for every complex polynomial p, where ‖p‖J denotes sup {|p(t)|:t ∈ J}. Such operators were introduced and first studied by Smart (4). They are of interest principally because they admit (and in fact are characterised by) an integral representation similar to, but in general weaker than, the integral representation of a self-adjoint operator on a Hilbert space. (See (2) and (4) for details.) It is easily seen, by verifying (1) directly, that T is well-bounded if it is a scalar-type spectral operator with real spectrum.

Foguel (8) and Fixman (7) independently proved that an invertible spectral operator, which is power-bounded, is of scalar type. Their proofs rely heavily on a result of Dunford on spectral operators whose resolvents satisfy a growth condition. (See Lemma 3.16 of (6, p. 609).) Observe that the resolvent of an invertible power-bounded operator T satisfies an inequality of the form