In this paper, the situation we shall be concerned with is that of a ring R, with a ring monomorphism α: R → R, which will not be assumed to be surjective.

Much work has been done on the skew polynomial ring R[x, α] and the skew Laurent polynomial ring R[x, x-1, α], where α is an automorphism—see [3] for example. However, the fact that α is not surjective renders the study of these objects much more difficult.

The aim of this work is to present a new approach to the concept of essential Fredholm complex of Banach spaces ([10], [2]; see also [11], [4], [6], [7] etc. for further connections), by using non-linear homogeneous mappings. We obtain some generalized homotopic properties of the class of essential Fredholm complexes, in our sense, which are then applied to establish its relationship with similar concepts. We also prove the stability of this class under small perturbations with respect to the gap topology.

An inverse semigroup S shall be said to be harmonic if every congruence on S is determined by any one of its classes. In other words, if λ and ρ are congruences on S having a congruence class in common, then λ = ρ. The class of all harmonic semigroups contains all bisimple inverse semigroups, as proved by Žitomirskiĭ [11] and also by Schein [10], and all congruence-free inverse semigroups. Moreover, is contained in the class of all 0-simple or simple inverse semigroups, as is easy to see. We shall show that there exist non-bisimple, non-congruence-free harmonic semigroups and that there are simple inverse semigroups which are not harmonic.

Various authors deal with distributive sublattices of the lattice ℒ(G) of subgroups of a group G. Perhaps the most basic result in this direction is due to O. Ore [9]: ℒ(G) is distributive if and only if G is locally cyclic.

In [11] and [12] finite groups with distributive lattices of subnormal subgroups were considered, while [3], [4], [7], [8], [10] and [13] deal with the case of groups G whose lattice N(G) of normal subgroups is distributive. Such groups were called DLN-groups in [10].

An arbitrary point of the Euclidean space Rn+1, where n > 1, is denoted by (X, y), where X ∈ Rn and y ∈ R, and we denote the Euclidean norm on Rn by ∥·∥. If h is harmonic on the half-space Ω = {(X, y): y > 0}, then we define extended real-valued functions m and M as follows:

and

Suppose that a group G is the semidirect product of a subgroup N and a normal subgroup M. Then the elements of G have unique expressions mn (m ∈ M, n ∈ N) and the commutator function

maps N x M into M. In fact there is an action (by automorphisms) of N on M given by

Conversely, if one is given an action of a group N on a group M then one can construct a semidirect product.

Free monoids play a central role in the theory of formal languages. Their endomorphisms appear naturally in the context of deterministic OL-schemes which trace their origin to biology. Closely related to such a scheme is a DOL-system which consists of a triple (X, φ, w) where X is a finite set, φ is an endomorphism of the free monoid X* and w ∈ X. The associated language is defined as the set {w, φw, φ2w,…} called a DOL-language. For a full discussion of this subject, we recommend the book [2] by Herman and Rozenberg.

In the literature there are several generalisations to non-commutative rings of the notion of a unique factorisation domain from commutative algebra. This paper follows in the spirit of [1, 3] and is set in the context of Noetherian rings. In [3], A. W. Chatters and the author denned a Noetherian UFR (unique factorisation ring) to be a prime Noetherian ring R in which every non-zero prime ideal contains a prime ideal generated by a non-zero normal element p, that is, by an element p such that pR = Rp. The class of Noetherian UFRs includes the Noetherian UFDs studied by Chatters in [1], while a commutative Noetherian ring is a UFR if and only if it is a UFD in the usual sense. For a Noetherian UFR R, the following are simple consequences of the definition:

(i) every non-zero ideal of R contains a non-zero normal element;

(ii) the set N(R) of non-zero normal elements of R is a unique factorisation monoid in the sense of [4, Chapter 3].

There is no example known of a non-nilpotent, torsion-free group which has all of its subgroups subnormal. It was proved in [3] that a torsion-free solvable group with all of its proper subgroups subnormal and nilpotent is itself nilpotent, but that seems to be the only published result in this area which is concerned specifically with torsion-free groups. Possibly the extra hypothesis that the group be hypercentral is sufficient to ensure nilpotency, though this is certainly not the case for groups with torsion, as was shown in [7]. The groups exhibited in that paper were seen to have hypercentral length ω + 1, and we know from [8] that further restricting the hypercentral length can lead to some positive results. Here we shall prove the following theorem.

A recent paper of Rhemtulla and Wilson [4] is concerned with elliptically embedded subgroups of groups. A subgroup H of a group G is elliptically embedded in G if, for each subgroup K of G, there is some integer m such that (H, K) = (HK)m. Some sufficient conditions for elliptic embedding are given in Section 2 of [4], and some consequences of the presence of this property are to be found in Theorems 1 and 2 of the same paper and in the main theorem of [5]. It is evident from all of these results that the property of being elliptically embedded is closely related to the nilpotency and subnormality of certain subgroups. One of the questions considered here i s the following.

We shall extend results of Samuel [19] and Griffin [8, 9] about conditions which generalise the notion of valuation domain in a field. Let U be a commutative ring with identity, R a subring of U and L an R-submodule of U. The conditions we study have in common the property (EV), that the submodules L:x (x ∈ U) form a chain. We pay particular attention to the strongest of the conditions, viz, that L be a Manis valuation (MV) subring, i.e. having a prime ideal P such that (L, P) is a maximal pair in U (see [19], [16] and e.g. [4]). Such P is unique, being the union of all L:x such that x ∉ L, which we call P+(L) the centre of L. This set P+ plays a key role in the study of all our valuation conditions.

A one-relator product Gof groups A and Bis defined to be the quotient of their free product A * B by the normal closure, «W»A*B, of a single element W, which is assumed to be cyclically reduced and of length at least 2. For convenience, the group Gwill be denoted by (A * B)/W.

Given a commutative ring A with identity, let W–1(A) denote the Witt group of skew-symmetric bilinear forms over A (cf. [1] or [7] for the definition of W–1 (A)).

In [8], [9] R. Rautmann has given a systematic development of error estimates for the spectral Galerkin approximations of the solution of the Navier–Stokes equations (spectral in the sense that one chooses as basis functions the eigenfunctions of the Stokes operator).

Error estimates are presented locally in time, valid on a finite interval determined by certain norms of the data. If one assumes the solution to be approximated is uniformly regular fort ≥ 0, the method gives an error estimate which grows exponentially with time. Without further assumptions this is the best estimate that one can expect. However, as pointed out in [2] by J. G. Heywood, if one assumes, additionally, that the solution to be approximated is stable, then one obtains an error estimate which is uniform in time.

In this note, we give an elementary procedure for constructing n-root closed integral domains. We then use this construction to give two interesting examples. First, we give an example of a root closed integral domain which is not quasinormal. Secondly, we show that for any subset 5 of odd positive primes there is a one-dimensional affine domain which is p-root closed for a prime p if and only if p ∈ S.

Let F be an orientable surface with or without boundary and let M(F) be the mapping class group of F, i.e. the group of isotopy classes of orientation preserving diffeomorphisms of F. To each essential simple closed curve c on F we can associate an element C of M(F) called the Dehn twist about c. We refer the reader to [1] for definitions. It is well known (see [1]) that, at least in the case where F has no more than one boundary component, M(F) is generated by Dehn twists. Further, there are important subgroups of M(F) which are also generated by Dehn twists or simple products of Dehn twists; for example the Torelli group, the kernel of the homology action map M(F)→ Aut(H1(F;Z)) = Sp(H1(F;Z)), where Sp(H1(F;Z)) denotes the symplectic group, is known to be generated by Dehn twists about bounding curves and by “bounding pairs”. See [8] for proofs and definitions. Also Dehn twists crop up as geometric monodromy maps associated to Picard–Lefschetz vanishing cycles for plane curve singularities (see [5]).