We give the first examples of contact metric spaces which are weakly locally $\varphi$-symmetric (that is, $\nabla R=0$ on horizontal vectors), but not strongly (that is, not all reflections with respect to the characteristic lines are local isometries). These examples are three-dimensional non-unimodular Lie groups with a left-invariant contact metric structure. We exhibit additional symmetries on these spaces.

1991 Mathematics Subject Classification: 53B20, 53C15, 53C25.

We show that the endomorphism rings of kernels ker ϕ of non-injective morphisms ϕ between indecomposable injective modules are either local or have two maximal ideals, the module ker ϕ is determined up to isomorphism by two invariants called monogeny class and upper part, and a weak form of the Krull–Schmidt theorem holds for direct sums of these kernels. We prove with an example that our pathological decompositions actually take place. We show that a direct sum of n kernels of morphisms between injective indecomposable modules can have exactly n! pairwise non-isomorphic direct-sum decompositions into kernels of morphisms of the same type. If ER is an injective indecomposable module and S is its endomorphism ring, the duality Hom(−, ER) transforms kernels of morphisms ER → ER into cyclically presented left modules over the local ring S, sending the monogeny class into the epigeny class and the upper part into the lower part.

We study the topological properties of cohomogeneity one flat manifolds and their orbits. Among other results we prove that principal orbits of R^n are isometric to R^{n-1} or S^k(c)\times R^{n-k-1}. We show that if M has one singular orbit, it is a totally geodesic submanifold of M and if M is orientable then there is at most one singular orbit.

The radical of a module over a commutative ring is the intersection of all prime submodules. It is proved that if R is a commutative domain which is either Noetherian or a UFD then R is one-dimensional if and only if every (finitely generated) primary R-module has prime radical, and this holds precisely when every (finitely generated) R-module satisfies the radical formula for primary submodules.

Let G be a finite group, (ZG) the group of units of the integral group ring ZG and 1(ZG) the subgroup of units of augmentation 1. In this paper, we are primarily concerned with the problem of describing constructively (ZG) for particular groups G. This has been done for a small number of groups (see [11] for an excellent survey), and most recently Jespers and Leal [3] described (ZG) for several 2-groups. While the situation is clear for all groups of order less than 16, not all groups of order 16 were discussed in their paper. Our main aim is to complete the description of (ZG) for all groups of order 16. Since the structure of the unit group of abelian groups is very well known (see for example [10]), we are only interested in the non-abelian case.

Let Q(x) = Q(x1, …, xn)∈ℤ[x1, …, xn] be a quadratic form. We investigate the size of the smallest non-zero solution of the congruence Q(x)≡0 (mod q). We seek a bound Bn(q), independent of Q, such that there is always a non-zero solution satisfying

The form gives the trivial lower bound Bn(q)≥(q/n)½ for all q and n, since if x≠0 and q∣ Q(x), then Q(x)≥q.

Let P be a prime ideal of a ring R, O(P) = {a ∊ R | aRs = 0, for some s ∊ R/P} | and Ō(P) = {x ∊ R | xn ∊ O(P), for some positive integer n}. Several authors have obtained sheaf representations of rings whose stalks are of the form R/O(P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P= Ō(P). In this paper we derive various conditions which ensure that a prime ideal P = Ō(P). The property that P = Ō(P) is then used to obtain conditions which determine when R/O(P) has a unique minimal prime ideal. Various generalizations of O(P) and Ō(P) are considered. Examples are provided to illustrate and delimit our results.

The modular group Г is the group of integral bilinear transformations of the extended complex plane which preserve the upper half-plane. It has the presentation 〈x, y:x2 = y3 = 1〉, and the generators can be chosen so that u = xy maps z to z + 1.

We consider higher secant varieties to Veronese varieties. Most points on the rth secant variety are represented by a finite scheme of length r contained in the Veronese variety – in fact, for a general point, the scheme is just a union of r distinct points. A modern way to phrase it is: the smoothable rank is equal to the border rank for most polynomials. This property is very useful for studying secant varieties, especially, whenever the smoothable rank is equal to the border rank for all points of the secant variety in question. In this note, we investigate those special points for which the smoothable rank is not equal to the border rank. In particular, we show an explicit example of a cubic in five variables with border rank 5 and smoothable rank 6. We also prove that all cubics in at most four variables have the smoothable rank equal to the border rank.

1. For any positive integral n and any positive real x1,…,xn we write

clearly. It is known [1,2] that

for n ≦ 6, and further [4, 5, 6] that (5) is false for even n ≧ 14 and for odd n ≧ 53. Mordell [2] conjectured that (5) is false for all n ≧ 7, but recently [3] stated that computations indicated that (5) is true for n = 7 and gave some calculations in support of (5) for n = 7.

Let f(n) = an2+ bn + c be an irreducible quadratic polynomial with integer coefficients, and let D denote the discriminant b2 – 4ac of f(n).We shall assume that (D, k) = 1, and that for all positive integer n, f(n) is positive and coprime with k, where k is a fixed integer greater than 1.

An inverse semigroup is called E-unitary if the equations ea = e = e2 together imply a2 = a. In a previous paper [4], the author showed that any E-unitary inverse semigroup is isomorphic to a semigroup constructed from a triple (G, ℋ, ) consisting of a down-directed partially ordered set ℋ, an ideal and subsemilattice of ℋ and a group G acting on ℋ, on the left, by order automorphisms in such a way that ℋ = G. This semigroup is denoted by P(G, ℋ, ); it consists of all pairs (a, g)∈ × G such that g−1a ∈ , under the multiplication

A natural class of ideals, almost isometric ideals, of Banach spaces is defined and studied. The motivation for working with this class of subspaces is our observation that they inherit diameter 2 properties and the Daugavet property. Lindenstrauss spaces are known to be the class of Banach spaces that are ideals in every superspace; we show that being an almost isometric ideal in every superspace characterizes the class of Gurariy spaces.

In this paper we prove the following:

Let S be a regular semigroup anda, bany elements of S such that Jb = ≦ Ja. Then, for each idempotent e∈Ja, there exists an idempotent f∈ Jb such that f = ≦e.

1. Let X be a closed Riemann surface of genus g ≥ 2 and let Aut X denote the group of automorphisms of X where, in this paper, an automorphism means a conformal or anticonformal self-homeomorphism. X is called hyperelliptic if it admits a conformal automorphism J of order 2 such that X/H has genus 0, where H = 〈J〉 is the group of order 2 generated by J. Thus X is a two-sheeted covering of the sphere which is branched over 2g + 2 points and J is the sheet-interchange map. J is the unique conformal automorphism of order 2 such that X/〈J〉 has genus 0 and it follows that if U ∈ Aut X, then UJU−1 = J. Thus J is central in Aut X and H ≤ Aut X. (Cf. [8])

Let Σg denote a compact orientable surface of genus g ≥ 2. We consider finite groups G acting effectively on Σg and preserving the orientation—for short, G acts on Σg or Gis a symmetry group of Σg. Each surface Σg admits only finitely many symmetry groups G and the orders of these groups are bounded by Wiman's bound of 84(g – 1). This bound is attained for infinitely many values of g [12], see also [9], and all values of g ≤ 104 for which it is attained are known [4].

Apart from simple Ore extensions such as the Weyl algebras, the best known example of a primitive Ore extension is the universal enveloping algebra U(g) of the 2-dimensional solvable Lie algebra g over a field k of characteristic zero, see [4, p. 22]. U(g) is a polynomial algebra over k in two indeterminates x and y with multiplication subject to the relation xy – yx = y, and may be regarded either as an Ore extension of k [x] by the k-automorphism which maps x to x – 1 or as an Ore extension of k[y] by the derivation yd/dy. The argument suggested in [4, p. 22] to prove the primitivity of U(g) can easily be generalised [6] to show that, if α is an automorphism of the ring R then the following conditions are sufficient for R[x, α] to be primitive: (i) no power αs, s ≧ 1, of α is inner; (ii) the only ideals of R invariant under α are 0 and R. These conditions are necessary and sufficient for the simplicity of the skew Laurent polynomial ring R[x, x–1, α] but are not necessary for the primitivity of R[x, α] (the ordinary polynomial ring D[x] over a division ring D not algebraic over its centre is easily seen to be primitive).

In this note we prove that a locally graded group $G$ in which all proper subgroups are (nilpotent of class not exceeding $n$)-by-Černikov, is itself (nilpotent of class not exceeding $n$)-by-Černikov.

As a preparatory result that is used for the proof of the former statement in the case of a periodic group, we also prove that a group $G$, containing a nilpotent of class $n$ subgroup of finite index, also contains a characteristic subgroup of finite index that is nilpotent of class not exceeding $n$.

Throughout the paper, rings are associative rings with identity. A ring is called right duo if every right ideal is two-sided, and it is called right p.p. if every principal right ideal is projective. A left duo (p.p.) ring is denned similarly, and a duo (p.p.) ring will mean a ring which is both right and left duo (p.p.). There is a right p.p. ring that is not left p.p. (see Chase [2[). Small [9] proved that right p.p. implies left p.p. if there are no infinite sets of orthogonal idempotents, and Endo [5, Proposition 2] has shown the same implication in the case where each idempotent in the ring is central. Since Courter [3, Theorem 1.3] noted that every idempotent in a right duo ring is central, we can simply speak of right duo p.p. rings. A typical example of a right duo ring which is not left duo is the following. Let F be a field and F(x) the field of rational functions over F. Let R = F(x)× F(x) as an additive group and define the multiplication as follows:

Then R is a local artinian ring with c(RR) = 2 and c(RR)= 3. Thus R is right duo but not left due.

Several definitions of a semigroup of quotients have been proposed and studied by a number of authors. For a survey, the reader may consult Weinert's paper [8]. The motivation for many of these concepts comes from ring theory and the various notions of rings of quotients. We are concerned in this paper with an analogue of the classical ring of quotients, introduced by Fountain and Petrich in [3].