Throughout this paper the word “ring” will mean an associative ring which need not have an identity element. There are Artinian rings which are not Noetherian, for example C(p∞) with zero multiplication. These are the only such rings in that an Artinian ring R is Noetherian if and only if R contains no subgroups of type C(p∞) [1, p. 285]. However, a certain class of Artinian rings is Noetherian. A famous theorem of C. Hopkins states that an Artinian ring with an identity element is Noetherian [3, p. 69]. The proofs of these theorems involve the method of “factoring through the nilpotent Jacobson radical of the ring”. In this paper we state necessary and sufficient conditions for an Artinian ring (and an Artinian module) to be Noetherian. Our proof avoids the concept of the Jacobson radical and depends primarily upon the concept of the length of a composition series. As a corollary we obtain the result of Hopkins.

A ring A has finite norm properly, abbreviated FNP, if each proper homomorphic image of A is finite. In [3], Chew and Lawn described some of the structural properties of FNP rings with identity, which they called residually finite rings. The twofold aim of this paper is to extend the results of [3] to arbitrary rings with FNP and to give characterizations of FNP rings independent of the results of [3].

If A is a ring, let A+ denote A regarded as an abelian group. In the first section of this paper, we explore the effects of FNP upon the structure of A+. The following theorem is typical of the results in this section.

The purpose of this paper is to characterize the radical ideals of principal ideal domains and Dedekind domains. We show that if T is a radical class and R is a PID, then T(R) is an intersection of prime ideals of R. More specifically, if

then T(R) = (p1p2 … pk), where p1, p2, … , pk are distinct primes, and where (p1p2 … Pk) denotes the principal ideal of R generated by p1p2 … pk. We also characterize the radical ideals of commutative principal ideal rings. For radical ideals of Dedekind domains we obtain a characterization similar to the one given for PID's.

The purpose of this paper is to exploit an analogy between algebraic extensions of fields and essential extensions of modules, in which the role of the algebraic closure of a field F is played by the injective hull H(M) of a unitary left R-module M. (The notion of * ‘algebraic’ extensions of general algebraic systems has been studied by Shoda; see, for example [5].)

In this analogy, the role of a polynomial p(x) is played by a homomorphism of R-modules

(1)

which will be called an ideal homomorphism into M. The process of solving the equation p(x) = 0 in F, or in an algebraic extension of F, will be replaced by the process of extending an ideal homomorphism (1) to a homomorphism F* from R into M, or into an essential extension of M.

In this paper we attempt to define the notion of normed Lie algebra by endowing the corresponding algebraic concept with topologicalmetric properties. More precisely, we define normed Lie algebras as being normed spaces possessing a Lie product, the latter satisfying a compatibility relation. It turns out that any normed algebra, in particular the algebra of continuous linear operators on a normed space, is a normed Lie algebra in the sense denned below, with the usual Lie product given by the additive commutator.

It seems that some algebraic features have within the framework of normed Lie algebras the natural topological extensions. We mention, for instance, the convergence of the well-known Campbell-Hausdorff formula. Let us also mention the occurence of a variant of the Kleinecke-Sirokov theorem, obtained via universal enveloping algebra, which might be unknown in this context.

Let K be a field with an involution J. A *-algebra over K is an associative algebra A with an involution * satisfying (α.a)* = αJ.a*. A large class of examples may be obtained as follows. Let (V, φ) be an hermitian space over K consisting of a vector space V and a left hermitian (w.r.t. J) form φ on V which is nondegenerate in the sense that φ(V,v) = 0 implies v = 0. An endomorphism f of V may have an adjoint f* w.r.t. φ, defined by φ(f(u),v) = φ(u,f*(v)); due to the nondegeneracy of φ, f* is unique if it exists. The set B(V, φ) of all endomorphisms of V which do have an adjoint is easily verified to be a *-algebra.

Let X be a topological space and Y a nonempty subspace of X. Γ(X, Y) denotes the semigroup under composition of all closed self maps of X which carry Y into Y, and is referred to as a restrictive semigroup of closed functions. Similarly, S(X, Y) is the analogous semigroup of continuous selfmaps of X, and is referred to as a restrictive semigroup of continuous functions. It is immediate that each homeomorphism from X onto U which carries the subspace Y of X onto the subspace V of U induces an isomorphism between Γ(X, Y) and Γ(U, V), and also an isomorphism between S(X, Y) and S(U, V). Indeed, one need only map f onto h o f o h-1. An isomorphism of this form is called representable. In [5, Theorem (3.1), p. 1223] it was shown that in most cases, each isomorphism from Γ(X, Y) onto Γ(U, V) is representable. The analogous problem was discussed for the semigroup S(X, Y) and it was pointed out by means of an example that one could not hope to obtain the same result for these semigroups without some further restrictions.

A Fuchsian group is a discrete subgroup of the hyperbolic group, L.F. (2, R), of linear fractional transformations

each such transformation mapping the complex upper half plane D into itself. If Γ is a Fuchsian group, the orbit space D/Γ has an analytic structure such that the projection map p: D → D/Γ, given by p(z) = Γz, is holomorphic and D/Γ is then a Riemann surface.

If N is a normal subgroup of a Fuchsian group Γ, then N is a Fuchsian group and S = D/N is a Riemann surface. The factor group, G = Γ/N, acts as a group of automorphisms (biholomorphic self-transformations) of S for, if γ ∊ Γ and z ∊ D, then γN ∊ G, Nz ∊ S, and (γN) (Nz) = Nγz. This is easily seen to be independent of the choice of γ in its N-coset and the choice of z in its N-orbit.

Conversely, if S is a compact Riemann surface, of genus at least two, then S can be identified with D/K, where K is a Fuchsian group acting without fixed points in D.

L. Fuchs has posed the problem of identifying those abelian groups that can serve as the additive structure of an injective module over some ring [1, p. 179], and in particular of identifying those abelian groups which are injective as modules over their endomorphism rings [1, p. 112]. Richman and Walker have recently answered the latter question, generalized in a non-trivial way [7], and have shown that the groups in question are of a rather restricted structure.

In this paper we consider abelian groups which are quasi-injective over their endomorphism rings. We show that divisible groups are quasi-injective as are direct sums of cyclic p-groups. Quasi-injectivity of certain direct sums (products) is characterized in terms of the summands (factors). In general it seems that the answer to the question of whether or not a group G is quasinjective over its endomorphism ring E depends on how big HomE(H, G) is, with H a fully invariant subgroup of G.

Let X be a metric space. Assume either that X is locally compact or that X has no more than countably many isolated points. It is proved that if F is a nowhere dense subset of X, then it is regularly nowhere dense (in the sense of Katětov) and hence is contained in the topological boundary of some regular-closed subset of X. This result is used to obtain new properties of the remote points of the Stone-Čech compactification of a metric space without isolated points.

Let βX denote the Stone-Čech compactification of the completely regular Hausdorff space X. Fine and Gillman [3] define a point p of βX to be remote if p is not in the βX-closure of a discrete subset of X.

A multifunction φ : X → Y from a topological space X into a topological space Y is a correspondence such that φ(x) is a non-empty subset of Y for every x ∊ X. A single-valued function f : X → Y is called a selection of φ if f(x) ∊ φ(x) for all x ∊ X; it is called a continuous selection if f is continuous. It is well-known that not every semi-continuous or even continuous multifunction has a continuous selection (see e.g. [4] for a survey on selection theory).

We investigate here some connections between multifunctions which are 'almost single-valued” and selections which are ‘almost continuous”.

The associated sequence space S of a sequence of vectors {xn} in a Banach space consists of all scalar sequences (sn) for which converges. My primary motivation in writing this paper was to present a new proof to a recent theorem of N. I. and V. I. Gurarii concerning limits of extent on S when {xn} is a basis of a uniformly convex or a uniformly smooth Banach space [5], This theorem is stated as Theorem 2.4. Several interesting consequences of this theorem were noted by N. I. Gurarii in [3] and [4].

This paper concerns a certain subalgebra of the Banach algebra of complex valued, essentially bounded, Lebesgue measurable functions on the unit circle in the complex plane (denoted here by L∞). My interest in this subalgebra was prompted by a question of R. G. Douglas. Let H∞ denote the space of functions in L∞ whose Fourier coefficients with negative indices vanish (equivalently, the space of boundary functions for bounded analytic functions in the unit disk). Douglas [5] has asked whether every closed subalgebra of L∞ containing H∞ is determined by the functions in H∞ that it makes invertible. More precisely, is such an algebra generated by H∞ and the inverses of the functions in H∞ that are invertible in the algebra? An affirmative answer is known for L∞ itself and for certain subalgebras of L∞ recently studied by Davie, Gamelin, and Garnett [3]. At the time of this writing, no algebra is known for which the above question can be answered negatively.

Let C be an n-square Hermitian matrix, presented in partitioned form as

where A is a-square and B is b-square. Let denote the eigenvalues of C, A, B, respectively. In a recent paper [10] the following inequality was established:

1.1

if

1.2

This inequality is a simplification and a sharpening of an inequality established earlier in [6], and is a wide generalization of an inequality of Aronszajn [4].

The purpose of this paper is to outline a simple theory of separability for a non-associative algebra A with semi-linear homomorphism σ. Taking A to be a finite dimensional abelian Lie p-algebra L and σ to be the pth power operation in L, this separability is the separability of [2]. Taking A to be an algebraic field extension K over k and σ to be the Frobenius (pth power) homomorphism in K, this separability is the usual separability of K over k. The theory also applies to any unital non-associative algebra A over a field k and any unital homomorphism σ from A to A such that σ(ke) ⊂ ke, e being the identity element of A.

In [6], J. Tits has shown that the Ree group 2F4(2) is not simple but possesses a simple subgroup of index 2. In this paper we prove the following theorem:

THEOREM. Let G be a finite group of even order and let z be an involution contained in G. Suppose H = CG(z) has the following properties:

(i) J = O2(H) has order 29and is of class at least 3.

(ii) H/J is isomorphic to the Frobenius group of order 20.

(iii) If P is a Sylow 5-subgroup of H, then Cj(P) ⊆ Z(J).

Then G = H • O(G) or G ≊ , the simple group of Tits, as defined in [6].

For the remainder of the paper, G will denote a finite group which satisfies the hypotheses of the theorem as well as G ≠ H • O(G). Thus Glauberman's theorem [1] can be applied to G and we have that 〈z〉 is not weakly closed in H (with respect to G). The other notation is standard (see [2], for example).

Let V be an n-dimensional vector space over the complex numbers equipped with an inner product (x, y), and let (P, μ) be a symmetry class in the mth tensor product of V associated with a permutation group G and a character χ (see below). Then for each T ∊ Hom (V, V) the function φ which sends each m-tuple (v1, … , vm) of elements of V to the tensor μ(TV1, … , Tvm) is symmetric with respect to G and x, and so there is a unique linear map K(T) from P to P such that φ = K(T)μ.

It is easily checked that K: Hom(V, V) → Hom(P, P) is a rational representation of the multiplicative semi-group in Hom(V, V): for any two linear operators S and T on V

K(ST) = K(S)K(T).

Moreover, if T is normal then, with respect to the inner product induced on P by the inner product on V (see below), K(T) is normal.

The intent of this paper is to study a class of algebras which do not necessarily obey the association law but instead obey a law which bears a marked resemblance to associativity. For lack of a better name we call this class the class of scalar dependent algebras. Specifically, an algebra A over a field F is called scalar dependent if there is a map g: A × A × A → F such that (xy)z = g(x, y, z)x(yz), for all x, y, z in A. To obtain our results we shall assume throughout that A is a scalar dependent algebra with an identity element e over a field of characteristic not 2 satisfying

(I) (x, x, x) = 0.

As usual, the associator (x,y,z) is defined by (x,y,z) = (xy)z — x(yz). An example is given to show that (I) is not implied by scalar dependency.

J. Lambek and G. Michler [3] have initiated the study of a ring of quotients RP associated with a two-sided prime ideal P in a right noetherian ring R. The ring RP is the quotient ring (in the sense of [1]) associated with the hereditary torsion class τ consisting of all right R-modules M for which HomR(M, ER(R/P)) = 0, where ER(X) is the injective hull of the R-module X.

In the present paper, we shall study further the properties of the ring RP. The main results are Theorems 4.3 and 4.6. Theorem 4.3 gives necessary and sufficient conditions for the torsion class associated with P to have property (T), as well as some properties of RP when these conditions are indeed satisfied, while Theorem 4.6 gives necessary and sufficient conditions for R to satisfy the right Ore condition with respect to (P).

A connected Lie group H is generated by a pair of one-parameter subgroups if every element of H can be written as a finite product of elements chosen alternately from the two one-parameter subgroups, i.e., if and only if the subalgebra generated by the corresponding pair of infinitesimal transformations is equal to the whole Lie algebra h of H (observe that the subgroup of all finite products is arcwise connected and hence, by Yamabe's theorem [5], is a sub-Lie group). If, moreover, there exists a positive integer n such that every element of H possesses such a representation of length at most n, then H is said to be uniformly finitely generated by the pair of one-parameter subgroups. In this case, define the order of generation of H as the least such n ; otherwise define it as infinity.