Much interest has been shown in determining the range of values of c for which the sequence [n]c contains infinitely many primes. The result is an elementary deduction from the prime number theorem, of course, of 0<c≤l. In 1953, Piatetski–Shapiro [9] showed that

for 1<c<12/11, where xc(X) stands for the number of primes in the set {[nc]n≤x}.

The Lindenstrauss formula

which states a strong relationship between the (Clarkson) modulus of uniform convexity δx of a Banach space X and the modulus of uniform smoothness px* of the conjugate space X*, is well known. Following the idea of the definitions of nearly uniform smooth space by S. Prus and modulus of uniform smoothness we define a modulus of nearly uniform smoothness and prove some Lindenstrauss type formulae concerning this modulus and the modulus of nearly uniform convexity for some measures of noncompactness.

All rings in this paper are associative but not necessarily with an identity. The ring R with an identity adjoined will be denoted by R#.

To denote that I is an ideal (right ideal, left ideal) of a ring R we write I ◃ R (I <rR, I <1R).

A ring R is called right (left) fully idempotent if for every I <rR (I<1R), I = I2.

At the conference “Methoden der Modul und Ringtheorie” in Oberwolfach, Germany in 1993, J. Clark raised the question as to whether every right fully idempotent ring is left fully idempotent (see also [3]). A similar question was raised by S. S. Page in [5]. In this note we answer the questions in the negative.

We start with some general observations most of which are perhaps well known. We include their simple proofs for completeness.

W. A. Bogley and M. A. Gutierrez [2] have recently obtained an eight-term exact homology sequence that relates the integral homology of a quotient group Г/MN, where M and N are normal subgroups of the group Г, to the integral homology of the free product Г/M * Г/N in dimensions ≤3 by means of connecting terms constructed from commutator subgroups of Г, M, N and M ∩ N. In this paper we use the methods of [4] to recover this exact sequence under weaker hypotheses and for coefficients in /q for any non-negative integer q. Further, for q = 0 we extend the sequence by three terms in order to capture the relation between the fourth homology groups.

M. E. Adams and Matthew Gould [1] have obtained a remarkable classification of ordered sets P for which the monoid End P of endomorphisms (i.e. isotone maps) is regular, in the sense that for every f є End P there exists g є End P such that fgf = f. They show that the class of such ordered sets consists precisely of

(a) all antichains;

(b) all quasi-complete chains;

(c) all complete bipartite ordered sets (i.e. given non-zero cardinals α β an ordered set Kα,β of height 1 having α minimal elements and β maximal elements, every minimal element being less than every maximal element);

(d) for a non-zero cardinal α the lattice Mα consisting of a smallest element 0, a biggest element 1, and α atoms;

(e) for non-zero cardinals α, β the ordered set Nα,β of height 1 having α minimal elements and β maximal elements in which there is a unique minimal element α0 below all maximal elements and a unique maximal element β0 above all minimal elements (and no further ordering);

(f) the six-element crown C6 with Hasse diagram

A similar characterisation, which coincides with the above for sets of height at most 2 but differs for chains, was obtained by A. Ya. Aizenshtat [2].

Let Fg be a closed orientable surface of genus g > 1 and let be the Teichmuller space of Fg, i.e., the space of marked hyperbolic structures on Fg We shall also denote by the space of marked hyperbolic structures on Fgwith one distinguished point; by this, we mean a distinguished point on the universal cover gof Fg. This space is isomorphic to the space of marked complete hyperbolic structures on a genus g surface with 1 cusp which is the usual interpretation of . Choose a decomposition of Fginto pairs of pants by a collection of non–intersecting, totally geodesic simple closed curves. The Fenchel–Nielsen coordinates for relative to this decomposition are given by the lengths of the curves as well as twist parameters defined on each curve. Varying the length and twist parameters gives deformations of the marked hyperbolic structures.

In this paper we refer to [13] and [16] for the basic terminology and properties of Noetherian rings. For example, an FBNring means a fully bounded Noetherian ring [13, p. 132], and a cliqueof a Noetherian ring Rmeans a connected component of the graph of links of R[13, p. 178]. For a ring Rand a right or left R–module Mwe use pr.dim.(M) and inj.dim.(M) to denote its projective dimension and injective dimension respectively. The right global dimension of Ris denoted by r.gl.dim.(R).

A number of classical theorems of ring theory deal with nilness and nilpotency of the Jacobson radical of various ring constructions (see [10], [18]). Several interesting results of this sort have appeared in the literature recently. In particular, it was proved in [1] that the Jacobson radical of every finitely generated PI-ring is nilpotent. For every commutative semigroup ring RS, it was shown in [11] that if J(R) is nil then J(RS) is nil. This result was generalized to all semigroup algebras satisfying polynomial identities in [15] (see [16, Chapter 21]). Further, it was proved in [12] that, for every normal band B, if J(R) is nilpotent, then J(RB) is nilpotent. A similar result for special band-graded rings was established in [13, Section 6]. Analogous theorems concerning nilpotency and local nilpotency were proved in [2] for rings graded by finite and locally finite semigroups.

A Banach space sE has the Grothendieck property if every (linear bounded) operator from E into c0 is weakly compact. It is proved that, for an integer k > 1, every k-homogeneous polynomial from E into c0 is weakly compact if and only if the space (kE) of scalar valued polynomials on E is reflexive. This is equivalent to the symmetric A>fold projective tensor product of £(i.e., the predual of (kE)) having the Grothendieck property. The Grothendieck property of the projective tensor product EF is also characterized. Moreover, the Grothendieck property of E is described in terms of sequences of polynomials. Finally, it is shown that if every operator from E into c0 is completely continuous, then so is every polynomial between these spaces.

In the nineteenth century, Hurwitz [8] and Wiman [14] obtained bounds for the order of the automorphism group and the order of each automorphism of an orientable and unbordered compact Klein surface (i. e., a compact Riemann surface) of topological genus g s 2. The corresponding results of bordered surfaces are due to May, [11], [12]. These may be considered as particular cases of the general problem of finding the minimum topological genus of a surface for which a given finite group G is a group of automorphisms. This problem was solved for cyclic and abelian G by Harvey [7] and Maclachlan [10], respectively, in the case of Riemann surfaces and by Bujalance [2], Hall [6] and Gromadzki [5] in the case of non-orientable and unbordered Klein surfaces. In dealing with bordered Klein surfaces, the algebraic genus—i. e., the topological genus of the canonical double covering, (see Alling-Greenleaf [1])—was minimized by Bujalance- Etayo-Gamboa-Martens [3] in the case where G is cyclic and by McCullough [13] in the abelian case.

Let M be an n-dimensional connected submanifold in an mdimensional Euclidean space Em. Denote by δ the Laplacian of M associated with the induced metric. Then the position vector x and the mean curvature vector H of Min Em satisfy

This yields the following fact: a submanifold M in Em is minimal if and only if all coordinate functions of Em, restricted to M, are harmonic functions. In other words, minimal submanifolds in Emare constructed from eigenfunctions of δ with one eigenvalue 0. By using (1. 1), T. Takahashi proved that minimal submanifolds of a hypersphere of Em are constructed from eigenfunctions of δ with one eigenvalue δ (≠0). In [3, 4], Chen initiated the study of submanifolds in Em which are constructed from harmonic functions and eigenfunctions of δ with a nonzero eigenvalue. The position vector x of such a submanifold admits the following simple spectral decomposition:

for some non-constant maps x0and xq, where A is a nonzero constant. He simply calls such a submanifold a submanifold of null 2-type.

In his celebrated paper [3] Gaschiitz proved that any finite non-cyclic p-group always admits non-inner automorphisms of order a power of p. In particular this implies that, if G is a finite nilpotent group of order bigger than 2, then Out (G) = Aut(G)/Inn(G) =≠1. Here, as usual, we denote by Aut (G) the full group of automorphisms of G while Inn (G) stands for the group of inner automorphisms, that is automorphisms induced by conjugation by elements of G. After Gaschiitz proved this result, the following question was raised: “if G is an infinite nilpotent group, is it always true that Out (G)≠1?”

Let S be a prime Noetherian ring and G a finite group acting on 5 such that Gis x-outer on S. We give sufficient conditions for the skew group ring S * Gto be a prime maximal order. If we impose the further hypothesis that the order of Gbe a unit of S, then these conditions are also necessary. Moreover, if S is a commutative Noetherian domain, then there are necessary and sufficient conditions for S*Gto be a prime maximal order, without requiring that the order of G be a unit in S.