Given an arbitrary associative ring R we consider the ring R[x] of polynomials over R in the commutative indeterminate x. For each radical property S we define the function S* which assigns to each ring R the ideal

of R. It is shown that the property SA (that a ring R be equal to S*(R)) is a radical property. If S is semiprime, then SA is semiprime also. If S is a special radical, then SA is a special radical. SA is always contained in S. A necessary and sufficient condition that S and SA coincide is given.

The results are generalized in the last section to include extensions of R other than R[x], One such extension is the semigroup ring R[A], where A is a semigroup with an identity adjoined.

All rings considered here are commutative containing at least two elements, but may not have identity. A ring R is said to be selfinjective if R as an R-module is injective. A ring R is said to be pre-selfinjective if every proper homomorphic image of R is self-injective [9]. Study of pre-self-injective rings was initiated by Levy [10], who established a characterization of Noetherian pre-self-injective rings with identity in terms of other well-known types of rings. Recently Klatt and Levy [9] have characterized all pre-self-injective rings with identity. In this paper we are mainly interested in Noetherian rings. For the sake of convenience we shall call a pre-self-injective ring an (I)-ring. A ring R will be said to be a (PMI)-ring if for each proper prime ideal P with P2 ≠ 0, the ring R/P2 is self-injective. Clearly, an (I)-ring is a (PMI)-ring.

In this paper all rings considered have identity and are commutative, and all modules are finitely generated. We shall make liberal use of the definitions and notation established in [6; 7].

Towber observed in [9] that a local Outer Product ring (OP-ring) must have v-dimension ≦ 2, and so a local OP-ring is either regular of global dimension ≦ 2 or it has infinite global dimension. Since the global dimension of a noetherian ring is the supremum of the global dimensions of its localizations, we immediately obtain the following result.

THEOREM 1.1. The global dimension of a noetherian OP-ring is either∞ or ≦ 2.

Let G be a p-primary Abelian group. Recall that the final rank of G is infn∈ω{r(pnG)}, where r(pnG) is the rank of pnG and ω is the first limit ordinal. Alternately, if Γ is the set of all basic subgroups of G, we may define the final rank of G by supB∈Γ {r(G/B)}. In fact, it is known that there exists a basic subgroup B of G such that r(G/B) is equal to the final rank of G. Since the final rank of G is equal to the final rank of a high subgroup of G plus the rank of pωG, one could obtain the same information if the definition of final rank were restricted to the class of p-primary Abelian groups of length ω.

Une structure de contact sur une variété M est un champ d'éléments de contact tangents Σ dont le système caractéristique Δ est régulier. Lorsque codim Σ = 1 une telle structure est définie localement par une forme de Pfaff ω ou encore par l'équation ω = 0. Le théorème classique de Darboux nous donne un critère d'équivalence de deux Pfaffiens ω et ω′ à l'aide de l'invariant classe ω. Cet invariant est cependant trop fort pour déterminer l'équivalence des structures de contact Σ et Σ′ définies par ω = 0 et ω′ = 0 respectivement. D'une façon précise, les structures Σ et Σ′ peuvent être équivalentes sans que les Pfaffiens ω et ω′ le soient. Dans cette note nous considérons un invariant plus faible, classe Σ, attaché à une structure de contact Σ et nous montrons qu'il détermine l'équivalence en codimension 1 (cf. corollaire 3).

Let cat be the Lusternik-Schnirelmann category structure as defined by Whitehead [6] and let be the category structure as defined by Ganea [2],

We prove that

and

It is known that w ∑ cat X = conil X for connected X. Dually, if X is simply connected,

1. We work in the category of based topological spaces with the based homotopy type of CW-complexes and based homotopy classes of maps. We do not distinguish between a map and its homotopy class. Constant maps are denoted by 0 and identity maps by 1.

We recall some notions from Peterson's theory of structures [5; 1] which unify the definitions of the numerical homotopy invariants akin to the Lusternik-Schnirelmann category.

In this paper we study algebraic closures for commutative semiprime rings. The main interest, however, is with rings which are regular in the sense of von Neumann. These play the same role with respect to semiprime rings as fields do with respect to integral domains. Two generally distinct notions are defined: “algebraic” and “weak-algebraic” extensions. Each has the transitivity property and yields a closure which is unique up to isomorphism and is “universal”. Both coincide in fields.

The extensions here called “algebraic” were studied independently by Enochs [5] and myself. Our results on these extensions proceed from a different point of view, and allow us to answer a question posed by Enochs. Furthermore, these results are required (and were developed) in order to obtain the weak-algebraic closure, which was the original closure sought. The motivation for the weak-algebraic extensions is found in the work of Shoda [14, p. 134, no. 1].

Local growth and decay estimates near the stationary point at the origin are derived in § 3 for solutions of the vector system,

(1)

where A(x) and B(y) are homogeneous of degree m > 1 in the components of x and y, respectively, and f* and g* are of order greater than m in ‖(x, y)‖ near the origin. It is assumed that x = 0 is asymptotically stable and y = 0 is asymptotically unstable for the homogeneous systems of first approximation,

(2)

In order to derive the estimates in § 3, various results are needed concerning solutions of a homogeneous system such as (2) (a). These are derived in § 2 and are based on work of Hahn [4; 5], Lefschetz [8], and Zubov [12].

In the study of compact semigroups the constructive method rather than the representational method is usually the better plan of attack. As it was pointed out by Hofmann and Mostert in the introduction to their book [10] this method is more productive than searching for a representation theory. Hofmann and Mostert described a constructive method called the Hormos and showed that any irreducible compact semigroup is obtained by the Hormos construction. Many of the important examples of irreducible semigroups which motivated their work were obtained by Hunter [11; 12; 13; 14].

In this paper, we apply the constructive method of ideal extensions [5] in algebraic semigroups to topological semigroups which are not necessarily compact. Many of Hunter's examples and examples of the Hormos technique can also be obtained by our method of topological ideal extensions. The topological ideal extension method, however, is, in general, a different type of construction technique.

All groups considered in this paper are abelian. A subgroup N of a group G is said to be a quasi-essential subgroup of G if G = 〈H, K〉 whenever H is an N-high subgroup of G and K is a pure subgroup of G containing N. We started the study of such subgroups in [5]; in particular, we characterized subsocles of a primary group which were both quasi-essential and centres of purity. In this paper we show that quasi-essential subsocles of a primary group are necessarily centres of purity answering thus in the affirmative a question raised in [5].

We obtain the following theorem: A subsocle S of a p-group G is quasi-essential if and only if either S ⊂ G1or (pnG)[p] ⊃ S ⊃ (pn+1G)[p] for some non-negative integer n. The notation is that of [1]. If G is a group, then

where p is a prime integer.

Let G be an additive group with nonempty subsets S and T. Let S ± T = {s ± t; s ∊ S, t ∊ T}, let S be the set complement of S in G, and let |S| be the cardinality of S. We abbreviate {f} where f ∊ G to f. If S + S and S have no element in common, then we say that S is a sum-free set in G or that S is sum-free in G. If S is a sum-free set in G and if for every sum-free set T in G, |S| ≧ |T|, then S is said to be a maximal sum-free set in G. We denote by λ(G) the cardinality of a maximal sum-free set in G. We say that S is in arithmetic progression having difference d if S = {s, s + d, …, s + nd} for some s, d ∊ G and some integer n ≧ 0.

That the class of fully complete (B-complete) locally convex spaces has an essential role has been well-established (see, e.g., Collins [2], Husain [12], and Pták [16]). However, it is still the case that very little is known concerning full-completeness in function spaces except when implied by formally stronger properties (e.g., Fréchet), and our aim in this article is to shed some light in this direction. Our approach is to use the unifying notion of the weighted spaces CV0(X) discussed in [19; 20], since such spaces provide a general setting for the study of virtually all continuous function spaces encountered in analysis.

The results to be presented here divide naturally into three sections with § 2 being consigned the preliminary definitions and theorems. In § 3 we obtain a necessary condition for full-completeness in a class of function spaces, and consider certain consequences of this result.

In [1, p. 41, Theorem 3.10] Arhangel'skiï proved that the perfect image of a completely regular space of countable type is of countable type, and he asked [1, p. 60, problem 4] if a similar result held for regular or Hausdorff spaces. In this paper, it is proved that the perfect image of a space of countable type is of countable type, provided that the image is Hausdorff or regular. An affirmative answer to both of Arhangel'skiï's questions follows immediately from this. Arhangel'skiï made use of the Stone-Čech compactification in the proof of his result, but the proofs below are of a different nature.

Let X be a topological space and let K ⊂ X. A collection of open sets is called a base at K provided that for every open set W ⊃ K there exists such that K ⊂ U ⊂ W. Clearly, we may assume that every member of contains K.

A famous theorem of Levitzki states that in a left Noetherian ring each nil left ideal is nilpotent. Lanski [5] has extended Levitzki's theorem by proving that in a left Goldie ring each nil subring is nilpotent. Another important theorem in this area which is due to Herstein and Small [3] states that if a ring satisfies the ascending chain condition on both left and right annihilators, then each nil subring is nilpotent. We give a short proof of a theorem (Theorem 1.6) which yields both Lanski's theorem and Herstein- Small's theorem. We make use of the ascending chain condition on principal left annihilators in order to obtain, at an intermediate step, a theorem (Theorem 1.1) which produces sufficient conditions for a nil subring to be left T-nilpotent.

Our aim in this paper is to continue our investigation of the homology of deleted products of finite, contractible, 2-dimensional polyhedra. In [1], we observed that if X is such a polyhedron, then a homeomorph of X can be constructed by starting with a 2-simplex and appending n-simplexes (n = 1, 2). In this paper, we are concerned with those polyhedra which have the property that if they are constructed as above, then at some stage we are forced to add to Xi–1 a 2-simplex τ at two of its 1-faces, 〈u3, u1〉 and 〈u3, u2〉, where there is a simple closed curve S in ∂(St(u3, Xi–1)) such that u1 and u2 are not in S but every sequence of 1-simplexes in ∂(St(u3, Xi–1)) from u1 to u2 intersects S. The cone over the complete graph on five vertices and the cone over the houses-and-wells figure are examples of such spaces.

Our aim in this paper is to extend to difference-differential rings the beautiful theorem of Kolchin [5, Theorem 3] for the differential case. The necessity portion of Kolchin's result is not obtained.

What might well be called the Ritt basis theorem states that if a commutative ring R with identity is finitely generated over a subring R0, then the ascending chain condition for radical ideals of R0 implies the ascending chain condition for radical ideals of R. (This is indeed a basis theorem. If we define a basis for a radical ideal A to be a finite set B such that then every radical ideal of a ring R has a basis if and only if the ascending chain condition for radical ideals holds in R.) It is the Ritt basis theorem rather than the Hilbert basis theorem which has appropriate generalizations in differential and difference algebra, where in fact it originated.

A Sturm-Liouville function is simply a non-trivial solution of the Sturm-Liouville differential equation

(1.1)

considered, together with everything else in this study, in the real domain. The associated quantities whose higher monotonicity properties are determined here are defined, for fixed λ > –1, to be

(1.2)

where y(x) is an arbitrary (non-trivial) solution of (1.1) and x1, x2, … is any finite or infinite sequence of consecutive zeros of any non-trivial solution z(x) of (1.1) which may or may not be linearly independent of y(x). The condition λ > –1 is required to assure convergence of the integral defining Mk, and the function W(x) is taken subject to the same restriction.

Let A denote the Banach algebra of functions analytic in the open unit disc D and continuous in . If f and its first m derivatives belong to A, then the boundary function f(eiθ) belongs to Cm(∂D). The space Am of all such functions is a Banach algebra with the topology induced by Cm(∂D). If all the derivatives of/ belong to A, then the boundary function belongs to C∞(∂D), and the space A∞ all such functions is a topological algebra with the topology induced by C∞(∂D). In this paper we determine the structure of the closed ideals of A∞ (Theorem 5.3).

Beurling and Rudin (see e.g. [7, pp. 82-89; 10]) have characterized the closed ideals of A, and their solution suggests a possible structure for the closed ideals of A∞.

Let S be a finite set and let S1, S2, …, St be subsets of S, not necessarily distinct. Does there exist a set of distinct representatives (SDR) for S1, S2, …, St? That is, does there exist a subset {a1, a2, …, at} of S such that ai ∊ Si, 1 ≦ i ≦ t, and ai ≠ aj if i ≠ j? The following theorem of Hall [2; 6, p. 48] gives the answer.

THEOREM. The subsets S1, S2, …, St have an SDR if and only if for each s, 1 ≦ s ≦ t, |Si1 ∪ Si1 ∪ … ∪ Sis| ≧ s for each s-comhination {i1, i2, …, is} of the integers 1, 2, …, t.

(Here and below, |A| denotes the number of elements in A.)