In this paper we study the ideal structure of the direct limit and direct sum (with a special multiplication) of a directed system of rings; this enables us to give descriptions of the prime ideals and radicals of semigroup rings and semigroup-graded rings.

We show that the ideals in the direct limit correspond to certain families of ideals from the original rings, with prime ideals corresponding to “prime” families. We then assume the indexing set is a semigroup ft with preorder defined by α≺β if β is in the ideal generated by α, and we use the direct sum to construct an Ω-graded ring; this construction generalizes the concept of a strong supplementary semilattice sum of rings. We show the prime ideals in this direct sum correspond to prime ideals in the direct limits taken over complements of prime ideals in Ω when two conditions are satisfied; one consequence is that when these conditions are satisfied, the prime ideals in the semigroup ring S[ft] correspond bijectively to pairs (Φ, Q) with Φ a prime ideal of Ω and Q a prime ideal of S. The two conditions are satisfied in many bands and in any commutative semigroup in which the powers of every element become stationary. However, we show that the above correspondence fails when Ω is an infinite free band, by showing that S[Ω] is prime whenever S is.

When Ω satisfies the above-mentioned conditions, or is an arbitrary band, we give a description of the radical of the direct sum of a system in terms of the radicals of the component rings for a class of radicals which includes the Jacobson radical and the upper nil radical. We do this by relating the semigroup-graded direct sum to a direct sum indexed by the largest semilattice quotient of Ω, and also to the direct product of the component rings.

We present a direct computation of the Rhodes expansions of the free objects in the varieties of bands, based in the manipulation of the invariants in introduced by Gerhard and Petrich [4] in the study of bands.

Suppose we are given a solid of revolution generated by a conic section. Slice out a frustum of the solid [14, diagrams pp. 77, 80]. Then, construct a cylinder, with the same height as the frustum, whose diameter coincides with the diameter of the frustum at the midpoint of its height. What is the difference between the volume of the frustum and the volume of this cylinder? Does this difference depend on where in the solid the frustum is taken?

The beautiful theorems which answer these questions first appear in a 1735 manuscript by Colin Maclaurin (1698–1746). This manuscript [14], the only original mathematical work by Maclaurin not previously printed, is published here for the first time, with the permission of the Trustees of the National Library of Scotland. (An almost identical copy [15] exists in the Edinburgh University Library.) In this work, Maclaurin proved that the difference between the cylinder constructed as above and the frustum of the given solid depends only on the height of the frustum, not the position of the frustum in the solid. When the solid is a cone, Maclaurin showed that its frustum exceeds the corresponding cylinder by one fourth the volume of a similar cone with the same height. For a sphere, the cylinder exceeds the frustum by one half the volume of the sphere whose diameter is equal to the height of the frustum; this holds, he observed, for all spheres. He derived analogous results for the ellipsoid and hyperboloid of revolution. Finally, for the paraboloid of revolution, he proved that the cylinder is precisely equal to the frustum.

Let L be a positive invertible self-adjoint operator in L2(X;C). Using transference methods for locally bounded groups of operators we obtain multipliers for the group of complex powers Liu on vector-valued Lebesgue spaces. Using a Mellin inversion formula, we derive a sufficient condition for a function to be a multiplier of the semigroup e-tL on Lp(X;E) where E is a UMD Banach space and 1<p<∞.

We introduce a class of regular extensions of regular semigroups, called enlargements; a regular semigroup T is said to be an enlargement of a regular subsemigroup S if S = STS and T = TST. We show that S and T have many properties in common, and that enlargements may be used to analyse a number of questions in regular semigroup theory.

If the group of inner automorphisms of a semigroup S of transformations of a finite n-element set contains an isomorphic copy of the alternating group Altn, then S is an Sn-normal semigroup and all the automorphisms of S are inner.

In this note we consider the Dirichlet problem Δu + f(x, u)=0 in Ω, u = 0 on ∂Ω here Ω is a bounded domain in ℝn(n≧3), with smooth boundary ∂Ω. We prove the existence of strong solutions to the previous problem, which are positive if f satisfies a suitable condition. As a consequence we find that the problem with , may have positive solutions even if g is not a lower-order perturbation of Next We examine the case .

It is shown that a ring R is right noetherian if and only if every cyclic right R-module is injective or a direct sum of a projective module and a noetherian module.

Rings of invariants are identified for some automorphisms θ of certain iterated skew polynomial rings R, including the enveloping algebra of sl2(k), the Weyl algebra A1 and their quantizations. We investigate how finite-dimensional simple R-modules split over the ring of invariants Rθ and how finite-dimensional simple Rθ-modules extend to R.

It is well known that a module M has finite length if and only if it is semi-artinian and Noetherian or, equivalently, semi-noetherian and artinian. Our main result shows that finite length is often achieved by just assuming that M is semi-artinian, semi-noetherian and has finitely generated socle.

We prove three results concerning the oscillation near a ray of solutions to (*)w″ + Aw = 0, where A is an entire function. The first result assumes that A is a polynomial and gives an upper bound on the number of its real zeros if (*) admits a solution with only real zeros and infinitely many. The second result proves that for A of finite order a solution w to (*) has “few” zeros “near” a ray if and only if the same is true for w′. The third result involves the density of the zeros of a solution to (*) “away” from a finite set of rays.

Saturated formations are closed under the product of subgroups which are connected by certain permutability properties.

We present some applications of monotonicity methods to the solution of certain nonscalar reaction-diffusion problems. In particular we prove existence under appropriate conditions and we introduce a convergent algorithm.

The construction of all irreducible modules of the symmetric groups over an arbitrary field which reduce to Specht modules in the case of fields of characteristic zero is given by G. D. James. Halicioğlu and Morris describe a possible extension of James' work for Weyl groups in general, where Young tableux are interpreted in terms of root systems. In this paper we show how to construct submodules of Specht modules for Weyl groups.

A Banach space X is called an HT space if the Hilbert transform is bounded from Lp(X) into Lp(X), where 1 < p < ∞. We introduce the notion of an ACF Banach space, that is, a Banach space X for which we have an abstract M. Riesz Theorem for conjugate functions in Lp(X), 1 < p < ∞. Berkson, Gillespie and Muhly [5] showed that X ∈ HT ⇒ X ∈ ACF. In this note, we will show that X ∈ ACF ⇒ X ∈ UMD, thus providing a new proof of Bourgain's result X ∈ HT ⇒ X ∈ UMD.

Let G be a finite group and denote by µ(G) (see [2]) the least positive integer m such that G has a faithful permutation representation in the symmetric group of degree m. This note considers the value of µ(G) when G is a double cover of the symmetric group.

We use Priestley's duality to characterize, via their dual space, the distributive double p-algebras on which all congruences are principal.

In this paper we generalize the folding process initiated by Stallings for graphs to a class of generalized covering spaces. These spaces are called pinched coverings or pinched cores, depending on the particular situation. We then apply our generalized folding process to manipulate these spaces into actual coverings. By using elementary homotopy arguments, we can calculate the fundamental groups of these spaces. As a corollary to our main result we obtain a generalization of a result due to Gersten concerning monomorphisms between free products of groups.

Existence results are established for the equation y″ + f(t, y) = 0, 0<t<1. Here f may be singular in y and f is allowed to change sign. Our boundary data include y(0) = y′(1) + ky(1) = 0, k> – 1 and y(0) = y′(1) + cy4(1) = 0, c>0.

An inverse transversal of a regular semigroup S is an inverse subsemigroup that contains precisely one inverse of each element of S. In the literature there are three known types of inverse transversal, namely those that are multiplicative, those that are weakly multiplicative, and those that form quasi-ideals. Here, by considering natural ways in which certain words can be simplified, we reveal four new types of inverse transversal. All of these can be illustrated nicely in examples that are based on 2 × 2 matrices.