We consider Fitting classes for which the injectors in any finite solvable group are modular subgroups. It is shown that only normal Fitting classes have this property. In fact, we prove two more general results demonstrating that modular Fitting functors and submodular Fitting classes are normal.

We give a description of factorisation in the ring of generalised power sums (the sequence of Taylor coefficients of rational functions regular at infinity) with a view to giving detailed bounds on the order of generalised power sum factors and roots of such sums.

A modified projection method is suggested for the approximate solution of second kind equations and it is compared with other methods.

Given a category A, we consider the (often large) set Ref A of its reflective (full, replete) subcategories, ordered by inclusion. It is known that, even when A is complete and cocomplete, wellpowered and cowellpowered, the intersection of two reflective subcategories need not be reflective. Supposing that A admits (i) small limits and (ii) arbitrary (even large) intersections of strong subobjects, we prove that an infimum ∧iCi in Ref A must necessarily be the intersection ∩iCi. Accordingly Ref A is not in general, even for good A, a complete lattice. We show, however, under the same conditions on A, that Ref A does admit small suprema ∨iCi, given by the closure in A of the union ∪iCi under the limits of type (i) and (ii) above.

All subsets of a field that are intersections of almost-orders of the field are characterised, and all ring topologies on a field which are not finer than any nontrivial locally bounded ring are characterised.

An example is given of an infinite cyclic extension of a free group of finite rank in which not every finitely generated subgroup is finitely separable. This answers negatively the question of Peter Scott as to whether in all finitely generated 3-manifold groups the finitely generated subgroups are finitely separable. In the positive direction it is shown that in knot groups and one-relator groups with centre, the finitely generated normal subgroups are finitely separable.

In this paper we prove a result concerning the Cauchy functional equation, that is the functional equation f(x + y) = f(x) + f(y), in skew fields with characteristic not two.

If H and M are right R-modules, H is M-injective if every R-homonorphism N → H, N a right R-submodule of M, can be extended to an R-homonorphism from M to H.

H is strongly M-injective if H is injective for inclusions whose cokernels are isomorphic to factor modules of M.

For the case of abelian groups H and M, one settles the questions “when is H M-injective” and “when is H strongly M-injective”. The latter can be characterized in terms of the vanishing of Ext. Results for general module categories are also given.

Let R be an integral domain with quotient field K. Ten conditions equivalent to “Either R is algebraic over ℤ or t.d.(R/Fp) ≤ 1 for some p” are given. One of these conditions, referred to in the title, is “Each extension of subrings of R having quotient field K satisfies the going-down property.” As consequences, other classes of rings are also characterised.

Let R denote the field of real numbers and let A be the ring of regular functions on Rn, that is the localization of R[T1 …, Tn] with respect to the set of all polynomials vanishing nowhere on Rn. Let X be an algebraic subset of Rn and let I(X) be the ideal of A of all functions vanishing on X. Assume that X is compact and nonsingular and k = codim X = 1, 2, 4 or 8. we prove here that if the A/I(X)-module I(X)/I(X)2 can be generated by k elements, then there exist a projective A-module P of rank k and a homomorphism from P onto I(X).