For any class of rings it is shown that the class (M) of all rings each nonzero homomorphic image of which contains either a nonzero -ideal or an essential ideal is a radical class. If is a class of simple rings the upper radical generated by , (M), is shown to be equal to (M) where ' is the class of simple rings complementary to .

This paper presents characterizations of optimality for the abstract convex program

when S is an arbitrary convex cone in a finite dimensional space, Ω is a convex set and p and g are respectively convex and S-convex (on Ω). These characterizations, which include a Lagrange multiplier theorem and do not presume any a priori constraint qualification, subsume those presently in the literature.

Reduced rings and lattice-ordered groups are examples of groups with Boolean orthogonalities. In this note we show that any group with a Boolean orthogonality satisfying a finiteness condition introduced by Stewart is isomorphic with a group of homeomorphisms of a topological space, in which two homeomorphisms are orthogonal if and only if they have disjoint supports.

An approach to p-adic interpolation via divided differences is used to give alternative proofs of results of van der Poorten on p-adic exponential polynomials and to derive a p-adic analogue of Turan's first main theorem on sums of powers.

Let G be a locally compact group G (which may be non-abelian) and Ap(G) the p-Fourier algebra of Herz (1971). This paper is concerned with the Fourier algebra Al, p(G) = Ap(G) ∩ L1(G) and various relations that exist between Al, p(G), Ap(G) and G.

Every poset with 0 is determined by various semigroups of isotone selfmaps which preserve 0. Two theorems along these lines are given and applied to some recent results concerning relation semigroups on topological spaces.

It is proved that (|τ(n)|n−11/2)δ has a mean-value for 0 <δ > < 2, where τ(n) is Ramanujan's function from modular arithmetic. Some further results are conjectured.

It is shown that if m, n are relatively prime positive integers, then the variety consisting of those soluble groups of exponent mn in which any subgroup of exponent m or n is abelian has a basis of two-variable laws.

A weak canonical form is derived for vector spaces of m × n matrices all of rank at most r. This shows that the structure of such spaces is controlled by the structure of an associated ‘primitive’ space. In the case of primitive spaces it is shown that m and n are bounded by functions of r and that these bounds are tight.

The iterative behaviour of polynomials is contrasted with that of small transcendental functions as regards the existence of unbounded domains of normality for the sequence of iterates.

Semiconvex sets are objects in the algebraic variety generated by convex subsets of real linear spaces. It is shown that the fundamental notions of convex geometry may be derived from an entirely algebraic approach, and that conceptual advantages result from applying notions derived from algebra, such as ideals, to convex sets. Some structural decomposition results for semiconvex sets are obtained. An algebraic proof of the algebraic Hahn-Banach theorem is presented.