1. Let X be a Banach space. Then a self-mapping A of X is said to be nonexpansive provided that ‖AX − Ay‖≤‖X − y‖ holds for all x, y ∈ X. The class of nonexpansive mappings includes contraction mappings and is properly contained in the class of all continuous mappings. Keeping in view the fixed point theorems known for contraction mappings (e.g. Banach Contraction Principle) and also for continuous mappings (e.g. those of Brouwer, Schauderand Tychonoff), it seems desirable to obtain fixed point theorems for nonexpansive mappings defined on subsets with conditions weaker than compactness and convexity. Hypotheses of compactness was relaxed byBrowder [2] and Kirk [9] whereas Dotson [3] was able to relax both convexity and compactness by using the notion of so-called star-shaped subsets of a Banach space. On the other hand, Goebel and Zlotkiewicz [5] observed that the same result of Browder [2] canbe extended to mappings with nonexpansive iterates. In [6], Goebel-Kirk-Shimi obtainedfixed point theorems for a new class of mappings which is much wider than those of nonexpansive mappings, and mappings studied by Kannan [8]. More recently, Shimi [12] used the fixed point theorem of Goebel-Kirk-Shimi [6] to discuss results for approximating fixed points in Banach spaces.

In 1956, Jacobson asked whether the intersection of the powers of the Jacobson radical, J(R), of a right Noetherian ring R, must always be zero [4, p. 200]. His question was answered in the negative by I. N. Herstein [3], who noted that , where Z(2) denotes the ring of rational numbers with denominator prime to 2, affords a counterexample. In contrast, the ring , though similar in appearance to R1, satisfies . (Here, k denotes a field.)

The purpose of this note is to prove the following result.

Theorem 1. Let n be an integer greater than zero. There exists a prime Noetherian ring R of Krull dimension n + 1 and a finitely generated essential extension W of a simple R-module V suchthat

(i) W has Krull dimension n, and

(ii) W/V is n-critical and cannot be embedded in any of its proper submodules.

We refer the reader to [6] for the definition and properties of Krull dimension.

Let G be a group. The rth term LrG of the lower central series of G is the subgroup generated by the r-fold commutators

where , and for r > 1,

Let ℤ and ℤ[i] have their usual meaning. Let Yo denote the noncommutative ring of integral quaternions, that is the set of all elements a + bi + cj + dk with a, b, c, d ∈ ℤ and where i, j and k together with the number 1 are the four units of the system of quaternions.

Recently there have appeared papers ([7], [8]; also see [9]) in which integral equations with kernels involving the confluent hypergeometric function

have been studied. These equations are mainly Volterra equations of the first kind except that they have infinite domain (0, ∞). The rest are of the related type with integrals over (x, ∞) instead of (0, x); and all are convolution equations.

Given a polynomially bounded multisequence {fm}, where m = (m1, …, mk) ∈ ℤk, we will consider 2k power series in exp(iz1), …, exp(izk), each representing a holomorphic function within its domain of convergence. We will consider this same multisequence as a linear functional on a class of functions defined on the k-dimensional torus by a Fourier series, , with the proper convergence criteria. We shall discuss the relationships that exist between the linear functional properties of the multisequence and the analytic continuation of the holomorphic functions. With this approach we show that a necessary and sufficient condition that the multisequence be given by a polynomial is that each of the power series represents, up to a unit factor, the same function that is entire in the variables

We say that a ring R has bounded index if there is a positive integer n such that an = 0 for each nilpotent element a of R. If n is the least such integer we say R has index n. For example, any semiprime right Goldie ring has bounded index, and so does any semiprime ring satisfying a polynomial identity [10, Theorem 10.8.2]. This paper is mainly concerned with the maximal (right) quotient ring Q of a semiprime ring R with bounded index. Several special cases of this situation have already received attention in the literature. If R satisfies a polynomial identity [1], or if every nonzero right ideal of R contains a nonzero idempotent [18] then it is known that Q is a finite direct product of matrix rings over strongly regular self-injective rings, the size of the matrices being bounded by the index of R. On the other hand if R is reduced (that is, has index 1) then Q is a direct product of a strongly regular self-injective ring and a biregular right self-injective ring of type III ([2] and [15]; the terminology is explained in [6]). We prove the following generalization of these results (see Theorems 9 and 11).

In this paper we shall indicate how to generalise the concept of a cofinite group (see [7]). We recall that any residually finite group can be made into a topological group by taking as a basis of neighbourhoods of the identity precisely the normal subgroups of finite index. The class of compact cofinite groups is then easily seen to be the class of profinite groups, where a group is profinite if and only if it is an inverse limit of finite groups. It turns out that every cofinite group can be embedded as a dense subgroup of a profinite group. This has important consequences for the class of countable locally finite-soluble groups with finite Sylow p-subgroups for all primes p, as shown in [7] and [14].

H will denote a Hilbert space of infinite dimension, ℬ(H) the algebra of bounded linear operators on H, and ℛ(H) the ideal of compact operators on H. We let σ, σe and σω denote the spectrum, essential spectrum and Weyl spectrum respectively. It is well known that for arbitrary T ∈ ℬ(H) we have by [5]

and

and

The aim of this paper is to discuss some relations among hereditary, strong and stable radicals. In particular we investigate hereditariness of lower strong and stable radicals. Some facts obtained are related to some results and questions of [2, 6, 7].

All rings in the paper are associative. Fundamental definitions and properties of radicals may be found in [9]. Definitions of hereditary and strong radicals are used as in Sands [7]. We say that a radical S is left (right) stable if

(ρ): for every ring R and every left (right) ideal I of R it follows S(I)⊆S(R).

Let H be a complex Hilbert space and denote by B(H) the Banach algebra of all bounded linear operators on H. In [5; 6] J. Ph. Labrousse proved that every operator S∈B(H) which is spectral in the sense of N. Dunford (see [3]) is similar to a T∈B(H) with the following property

Conversely, he showed that given an operator S∈B(H) such that its essential spectrum (in the sense of [5; 6]) consists of at most one point and such that S is similar to a T∈B(H) with the property (1), then S is a spectral operator. This led him to the conjecture that an operator S∈B(H) is spectral if and only if it is similar to a T∈B(H) with property (1). The purpose of this note is to prove this conjecture in the case of operators which are decomposable in the sense of C. Foias (see [2]).