Let denote the contracted semigroup ring of the ompletely 0-simple semigroup D over the ring R. The Rees structure theory of completely 0-simple semigroups is used to obtain necessary and sufficient conditions that have zero radical (Theorem 3.8). By using Amitsur's construction of the upper π-radical [1], we are able to treat the Jacobson, Baer (prime), Levitzki (locally nilpotent) and possibly the nil radicals simultaneously. Our results generalize a theorem of Munn [6] on semigroup algebras of finite 0-simple semigroups.

We gave an elementary proof of Theorem 4 in the paper, published in the Glasgow Mathematical Journal 39(1997), 275–284. The result provides an algorithm for approximating the maximal eigenvalue of a nonnegative matrix. Recently the author has learnt that the result can be proved immediately from Theorem 6.8 in [1]. Indeed, the paper [1] determines necessary and sufficient conditions for the convergence of an iterative sequence to the maximal eigenvalue. Their proof needs knowledge of graph theoretical concepts.

We begin this paper by considering a Boolean algebra as a lattice which is relatively pseudo-complemented (i.e., residuated with respect to intersection) and give, in this case, certain properties of the equivalences of types A, B and F(as introduced by Molinaro [1]). We then show how these results carry over to the case of Boolean matrices, which form a Boolean algebra residuated also with respect to matrix multiplication. Other properties of matrix residuals are established and we conclude with three algebraic characterisations of invertible Boolean matrices.

From the proof of Theorem 2 of [5] it follows that for every positive integer k there exist infinitely many primes p in the arithmetical progression ax + b (x = 0, 1, 2,…), where a and b are relatively prime positive integers, such that the number 2p−1 − 1 has at least k composite factors of the form (p − 1)x + 1. The following question arises:

For any given natural number k, do there exist infinitely many primes p such that the number 2p−1 − 1 has k prime factors of the form(p − 1)x + 1 and p ≡ b (mod a), where a and b are coprime positive integers?

In a semigroup S the set E of idempotents is partially ordered by the rule that e≦ƒ if and only if eƒ=e=ƒe. We say that S is an ω-semigroup if E={ei: i=0, 1, 2, …}, where

Bisimple ω-semigroups have been classified in [10]. From a group G and an endomorphism α of G a bisimple ω-semigroup S(G, α) can be constructed by a process described below in § 1: moreover, any bisimple ω-semigroup is isomorphic to one of this type.

We prove the pointwise inequality 0 ≥ ρ + ρ⊥ – 1 involving the normalized scalar curvature ρ and normal scalar curvature ρ⊥ of a totally real 3-dimensional submanifold of the nearly Kaehler 6-sphere. Further we classify submanifolds realizing the equality in this inequality.

Let A be a commutative Noetherian ring (with non-zero identity). The Cousin complex C(A) for A is described in [6, §2]: it is a complex of A-modules and A-homomorphisms

with the property that, for each n≥0,

Cohen-Macaulay rings may be characterized in terms of the Cousin complex: A is a Cohen-Macaulay ring if and only if C(A) is exact [6, (4.7)]. Also the Cousin complex provides a natural minimal injective resolution for a Gorenstein ring: see [6, (5.4)].

We denote by Autsn(G) the set of all automorphisms that fix every subnormal subgroup of G setwise. In a recent paper [7] Robinson showed that the structure of Autsn(G) is quite restricted for a finite group G. Our aim in this paper is to show that more detailed information about the structure of Autsn(G) can be obtained by focussing on its action on F*(G), the generalised Fitting subgroup of G.

Throughout the present note we abbreviate the set of p parameters a1,…,ap by (ap), with similar interpretations for (bq), etc. Also, by [(ap)]m we mean the product , where [λ]m = Г(λ + m)/ Г(λ), and so on. One of the main results we give here is the expansion formula

(1)

which is valid, by analytic continuation, when, p,q,r,s,t and u are nonnegative integers such that p+r < q+s+l (or p+r = q+s+l and |zω| <1), p+t < q+u (or p + t = q + u and |z| < 1), and the various parameters including μ are so restricted that each side of equation (1) has a meaning.

In [2] I described the deformation of bending a hyperbolic manifold along an embedded totally geodesic hypersurface. As I remarked there, the deformation is particularly interesting in the case of a surface, because a surface contains many embedded totally geodesic hypersurfaces, namely simple closed curves, along which it is possible to bend. Furthermore, for a surface it is possible to extend the definition of bending to the case of a geodesic lamination, by using the fact that the set of simple closed geodesies is dense in the space of geodesic laminations. This direction has been developed by Epstein and Marden in [1].

The Bass–Serre theorem supplies generators and relations for a group of automorphisms of a tree. Recently K. S. Brown [2] has extended the result to produce a presentation for a group of automorphisms of a simply connected complex, the extra ingredient being relations which come from the 2-cells of the complex. Suppose G is the group, K the complex and L the 1-skeleton of K. Then an extension of π1(L) by G acts on the universal covering space of L (which is of course a tree) and Brown's technique is to apply the work of Bass and Serre to this action. Our aim is to give a direct elementary proof of Brown's theorem which makes no use of covering spaces, deals with the Bass–Serre theorem as a special case and clarifies the roles played by the various generators and relations.

In a recent paper [1], the theory of non-linear semi-special permutations has been developed. A method for describing such permutations, on a given range [n], is to be found in §§2, 3. This method consists in choosing a proper divisor s of n, determining all the semispecial permutations with principal number s, and then making stake all its possible values. In this way the totality of non-linear semi-special permutations on [n] may be obtained.

All groups G considered in this paper are finite and all representations of G are defined over the field of complex numbers. The reader unfamiliar with projective representations is referred to [9] for basic definitions and elementary results.

Let Proj(G, α) denote the set of irreducible projective characters of a group G with cocycle α. In a previous paper [3] the author showed that if G is a (p, α)-group, that is the degrees of the elements of Proj(G, α) are all powers of a prime number p, then G is solvable. However Isaacs and Passman in [8] were able to give structural information about a group G for which ξ(1) divides pe for all ξ ∈ Proj(G, 1), where 1 denotes the trivial cocycle of G, and indeed classified all such groups in the case e = l. Their results rely on the fact that G has a normal abelian p-complement, which is false in general if G is a (p, α)-group; the alternating group A4 providing an easy counter-example for p = 2.

We show that Diestel's theorem on weak compactness of subsets of L1,(μ, X) can be derived as a simple corollary of James's theorem. It is a pleasure to acknowledge several stimulating conserversations with Dave Emmons and the remarks of an anonymous referee. Errors are, of course, solely mine.