A unital homomorphism f: R → T of commutative rings is said to be nearly integral if the induced map R/I → T/IT is integral for each ideal I of R which properly contains ker (f). This concept leads to new characterisations of integral extensions and fields. For instance, if R is not a field, then an inclusion R → T is integral if and only if it is nearly integral and (R, T) is a lying-over pair. It is also proved that each overring extension of an integral domain R is nearly integral if and only if dim (R) ≤ 1 and the integral closure of R is a Prüfer domain. Related properties and examples are also studied.

We define a certain class of groups, Ck, which we show to contain the class of all k-free groups. Our main theorem shows that certain amalgamated free products of groups in C3, are again in C3. In the appendix we show that many 3-manifold groups belong to Ck for suitable k.

We give a construction of 2s, s = n(n – 1)/2, many natural almost complex structures on the orthogonal twistor bundle over a 2n-dimensional Riemannian manifold. The usual almost complex structures are then characterised by the condition that they correspond to integrable invariant complex structures on the standard fibre which is identified with the hermitian symmetric space SO(2n)/U(n).

We study non-commutative convolution measure algebras satisfying the condition in the title and having an involution with a non-degenerate finite dimensional *-representation. We show first that the group algebra L1(G) of a locally compact group G satisfies these conditions. Then we show that to a given algebra A with the above conditions there corresponds a locally compact group G such that A is a * and L-subalgebra of M(G) and such that the enveloping C*-algebra of A is *isomorphic to C*(G). Finally we show for certain groups that L1(G) is the only example of such algebras, thus giving a characterisation of L1(G).

Sufficient or necessary conditions are established so that the neutral functional differential equation [x(t) − G(t, xt)]″ + F(t, xt) = 0 has a solution which is asymptotic to a given solution of the related difference equation x(t) = G(t, xt) + a + bt, where a and b are constants.

It is shown that the natural generalisations of the elementary Nielsen transformations of a free group to the infinite-rank case, furnish generators for the subgroup of “bounded” automorphisms of any relatively free nilpotent group of infinite rank. This settles the nilpotent analogue of a question of D. Solitar concerning the “bounded” automorphisms of absolutely free groups of infinite rank.

Precise decay estimates as t → ∞ are derived for a class of nonlinear second order ordinary differential equations of the form

where h, g and f are functions like

With α > −1, β > −1 and γ > −1.

The concept of the heredity measure of a semiprimary ring (or finite-dimensional algebra) is introduced and some of its elementary properties are studied.

We consider the following non-linear nonautonomous second order differential equation

where h(x) is continuous, f, p are continuous and periodic with respect to t of period w. Using the Leray-Schauder fixed point technique we prove that the above equation possesses at least one non-trivial periodic solution of period w.

We use the theory of clones to prove that a countably presented variety of algebras can be embedded in a variety of groupoids.

Let X = [a, b] be a closed bounded real interval. Let B be the closed linear space of all bounded real valued functions defined on X, and let M ⊆ B be the closed convex cone consisting of all monotone non-decreasing functions on X. For f, g ∈ B and a fixed positive w ∈ B, we define the so-called best L∞-simultaneous approximant of f and g to be an element h* ∈ M satisfying

for all h ∈ M, where

We establish a duality result involving the value of d in terms of f, g and w only.

If in addition f, g and w are continuous, then some characterisation results are obtained.

Let R be a ring with centre Z. In this note, we prove the following: If the additive group Z+ of Z has finite group-theoretic index in R+, then R has an ideal I contained in Z such that R/I is a finite ring. This is a solution of a problem posed by F.A. Szász.

For F free of rank 4, a generating set for Aut (F/[F′, F′, F′]) exhibiting countably many wild automorphisms is obtained. Also included are examples of wild automorphisms in Aut (F/Vk+1) which are tame in Aut (F/Vk) for k ≥ 1, where Vk = [F″, F,…, F] (F repeats k times).

Neumann's totally ordered power series fields and division rings may be tipped over to form archimedean lattice-ordered fields and division rings. This process is described and then generalised to produce non-archimedean lattice-ordered fields and division rings in which 1 > 0 and, as well, ones in which 1 ≯ 0.

It is shown that Witt's basic dimension formula and a more recent result of Klyachko imply each other. Then Klyachko's identities between certain idempotents in the group ring of Sn are supplemented by identities involving Wever's classical idempotent. This leads to a direct proof of Klyachko's theorem (and hence Witt's formula), avoiding any commutator collecting process. Furthermore, this approach explains why the Witt dimensions are numbers which otherwise occur when “counting necklaces”.

We present some new results on third Engel groups which are motivated by computer calculations but are not dependent on them. They include:

• for n > 2 every n-generator third Engel group is nilpotent of class at most 2n – 1;

• the fifth term of the lower central series of a third Engel group has exponent dividing 20;

• the subgroup generated by fifth powers of elements in a third Engel group is nearly centre-by-metabeliami;

and a normal form theorem for freest third Engel groups without elements of order 2.

A natural concept of variety for regular semigroups is introduced: an existence variety (or e-variety) of regular semigroups is a class of regular semigroups closed under the operations H, Se, P of taking all homomorphic images, regular subsernigroups and direct products respectively. Examples include the class of orthodox semigroups, the class of (regular) locally inverse semigroups and the class of regular E-solid semigroups. The lattice of e-varieties of regular semigroups includes the lattices of varieties of inverse semigroups and of completely regular semigroups. A Birkhoff-type theorem is proved, showing that each e-variety is determined by a set of identities: such identities are then given for many e-varieties. The concept is meaningful in universal algebra, and as for regular semigroups could give interesting results for e-varieties of regular rings.

Section 1 addresses the problem of covers in Sub D, the lattice of subalgebras of a Boolean algebra; we describe those BA's in whose subalgebra lattice every element has a cover, and show that every small and separable subalgebra of P(ω) has 2ω covers in SubP(ω). Section 2 is concerned with complements and quasicomplements. As a general result it is shown that Sub D is relatively complemented if and only if D is a finite– cofinite BA. Turning to Sub P(ω), we show that no small and separable D ≤ P(ω) can be a quasicomplement. In the final section, generalisations of packed algebras are discussed, and some properties of these classes are exhibited.

A class of blocking pairs (A, B) for which B/AB is infinite cyclic is constructed and these are used to construct groups all of whose proper verbal quotients are relatively free.

The purpose of this note is to describe some of the standard data types of computer science in the language of distributive categories. We believe that in this way we have achieved a simplification and a formal clarification of the specification of these data types.