Rings defined by $\mathcal{R}$-sets and a characterization of a class of semiperfect rings
Robert
Gordon
1-17
Completely $0$-simple semirings
Mireille Poinsignon
Grillet;
Pierre-Antoine
Grillet
19-33
Abstract: A completely $( - 0)$ simple semiring is a semiring $R$ which is $( - 0)$ simple and is the union of its $( - 0)$ minimal left ideals and the union of its $( - 0)$ minimal right ideals. Structure results are obtained for such semirings. First the multiplicative semigroup of $R$ is completely $( - 0)$ simple; for any $ \mathcal{H}$-class $H( \ne 0),H( \cup \{ 0\} )$ is a subsemiring. If furthermore $R$ has a zero but is not a division ring, and if $(H \cup \{ 0\} , + )$ has a completely simple kernel for some $H$ as above (for instance, if $R$ is compact or if the $\mathcal{H}$-classes are finite), then (i) $ (R, + )$ is idempotent; (ii) $R$ has no zero divisors, additively or multiplicatively. Additional results are given, concerning the additive $ \mathcal{J}$-classes of $ R$ and also $( - 0)$ minimal ideals of semirings in general.
Spheroidal decompostions of $E\sp{4}$
J. P.
Neuzil
35-64
Abstract: This paper investigates a generalization to ${E^4}$ of the notion of toroidal decomposition of $ {E^3}$. A certain type of this kind of upper semicontinuous decomposition is shown to be shrinkable and hence yield $ {E^4}$ as its decomposition space.
A generlization of Feit's theorem
J. H.
Lindsey
65-75
Abstract: This paper is part of a doctoral thesis at Harvard University. The title of the thesis is Finite linear groups in six variables. Using the methods of this paper, I believe that I can prove that if $p$ is a prime greater than five with $p \equiv - 1\pmod 4$, and $G$ is a finite group with faithful complex representation of degree smaller than $4p/3$ for $p > 7$ and degree smaller than 9 for $p = 7$, then $G$ has a normal $p$-subgroup of index in $G$ divisible at most by ${p^2}$. These methods are particularly effective when there is nontrivial intersection of $ p$-Sylow subgroups. In fact, if the current work people are doing on the trivial intersection case can be extended, it should be possible to show that, for $p$ a prime and $G$ a finite group with a faithful complex representation of degree less than $ 3(p - 1)/2,G$ has a normal $ p$-subgroup of index in $ G$ divisible at most by $ {p^2}$. (It may be possible to show that the index is divisible at most by $ p$ if the representation is primitive and has degree unequal to $p$.)
Stiefel-Whitney numbers of quaternionic and related manifolds
E. E.
Floyd
77-94
Abstract: There is considered the image of the symplectic cobordism ring $\Omega _\ast^{SP}$ in the unoriented cobordism ring $ {N_\ast }$. A polynomial subalgebra of ${N_\ast }$ is exhibited, with all generators in dimensions divisible by 16, such that the image is contained in the polynomial subalgebra. The methods combine the $K$-theory characteristic numbers as used by Stong with the use of the Landweber-Novikov ring.
Under the degree of some finite linear groups
Harvey I.
Blau
95-113
Abstract: Let $G$ be a finite group with a cyclic Sylow $ p$-subgroup $P$ for some prime $p \geqq 13$. Assume that $G$ is not of type ${L_2}(p)$, and that $G$ has a faithful indecomposable modular representation of degree $d \leqq p$. This paper offers several improvements of the known bound $ d \geqq (7p)/10 - 1/2$. In particular, $ d \geqq 3(p - 1)/4$. Other bounds are given relative to the order of the center of $G$ and the index of the centralizer of $ P$ in its normalizer.
On the injective hulls of semisimple modules
Jeffrey
Levine
115-126
Abstract: Let $R$ be a ring. Let $T = { \oplus _{i \in I}}E(R/{M_i})$ and $W = \prod\nolimits_{i \in I} {E(R/{M_i})}$, where each ${M_i}$ is a maximal right ideal and $E(A)$ is the injective hull of $ A$ for any $R$-module $A$. We show the following: If $R$ is (von Neumann) regular, $ E(T) = T$ iff ${\{ R/{M_i}\} _{i \in I}}$ contains only a finite number of nonisomorphic simple modules, each of which occurs only a finite number of times, or if it occurs an infinite number of times, it is finite dimensional over its endomorphism ring. Let $ R$ be a ring such that every cyclic $R$-module contains a simple. Let ${\{ R/{M_i}\} _{i \in I}}$ be a family of pairwise nonisomorphic simples. Then $E({ \oplus _{i \in I}}E(R/{M_i})) = \prod\nolimits_{i \in I} {E(R/{M_i})} $. In the commutative regular case these conditions are equivalent. Let $ R$ be a commutative ring. Then every intersection of maximal ideals can be written as an irredundant intersection of maximal ideals iff every cyclic of the form $ R/\bigcap\nolimits_{i \in I} {{M_i}}$, where ${\{ {M_i}\} _{i \in I}}$ is any collection of maximal ideals, contains a simple. We finally look at the relationship between a regular ring $R$ with central idempotents and the Zariski topology on spec $R$.
The family of all recursively enumerable classes of finite sets
T. G.
McLaughlin
127-136
Abstract: We prove that if $ P(x)$ is any first-order arithmetical predicate which enumerates the family Fin of all r.e. classes of finite sets, then $P(x)$ must reside in a level of the Kleene hierarchy at least as high as $\prod _3^0 - \Sigma _3^0$. (It is more easily established that some of the predicates $P(x)$ which enumerate Fin do lie in $\prod _3^0 - \Sigma _3^0$.)
Topologies for $2\sp{x}$; set-valued functions and their graphs
Louis J.
Billera
137-147
Abstract: We consider the problem of topologizing ${2^X}$, the set of all closed subsets of a topological space $X$, in such a way as to make continuous functions from a space $Y$ into ${2^X}$ precisely those functions with closed graphs. We show there is at most one topology with this property, and if $X$ is a regular space, the existence of such a topology implies that $X$ is locally compact. We then define the compact-open topology for ${2^X}$, which has the desired property for locally compact Hausdorff $X$. The space ${2^X}$ with this topology is shown to be homeomorphic to a space of continuous functions with the well-known compact-open topology. Finally, some additional properties of this topology are discussed.
Compact functors and their duals in categories of Banach spaces
Kenneth L.
Pothoven
149-159
Abstract: In a recent paper, B. S. Mityagin and A. S. Shvarts list many problems concerning functors and dual functors in categories of Banach spaces. Included in these problems is the question: What properties characterize compact functors? The purpose of this paper is to give partial answers to that question. Partial characterizations are given in terms of what are called Fredholm functors and finite rank functors. Affirmative answers are also given to two other questions of Mityagin and Shvarts. They are (1) If a functor is compact, is its dual compact? (2) If a natural transformation is compact, is its dual compact?
Jordan algebras with minimum condition
David L.
Morgan
161-173
Abstract: Let $J$ be a Jordan algebra with minimum condition on quadratic ideals over a field of characteristic not 2. We construct a maximal nil ideal $R$ of $J$ such that $J/R$ is a direct sum of a finite number of ideals each of which is a simple Jordan algebra. $R$ must have finite dimension if it is nilpotent and this is shown to be the case whenever $ J$ has ``enough'' connected primitive orthogonal idempotents.
Perturbations of solutions of Stieltjes integral equations
David Lowell
Lovelady
175-187
Abstract: Using multiplicative integration in two ways, formulae for solutions to perturbed Stieltjes integral equations are found in terms of unperturbed solutions. These formulae are used to obtain bounds on the difference between the perturbed solution and the unperturbed solution. The formulae are also used to explicitly solve, in terms of product integrals, a linear equation subject to nonlinear interface conditions.
Automorphisms of group extensions
Charles
Wells
189-194
Abstract: If $1 \to G\mathop \to ^\iota \Pi \mathop \to ^\eta 1$ is a group extension, with $ \iota$ an inclusion, any automorphism $\varphi$ of Let $\overline \alpha :\Pi \to $ Out $G$ be the homomorphism induced by the given extension. A pair $ (\sigma ,\tau ) \in {\rm {Aut }}\Pi \times {\rm {Aut }}G$ is called compatible if $\sigma$ fixes $ \ker \overline \alpha$, and the automorphism induced by $\sigma$ on $ \Pi \overline \alpha$ is the same as that induced by the inner automorphism of Out $G$ determined by $\tau$. Let $C < {\rm {Aut }}\Pi \times {\rm {Aut }}G$ be the group of compatible pairs. Let ${\rm {Aut (}}E;G{\rm {)}}$ denote the group of automorphisms of $E$ fixing $G$. The main result of this paper is the construction of an exact sequence $\displaystyle 1 \to Z_\alpha ^1(\Pi ,ZG) \to \operatorname{Aut} (E;G) \to C \to H_\alpha ^2(\Pi ,ZG).$ The last map is not surjective in general. It is not even a group homomorphism, but the sequence is nevertheless ``exact'' at $C$ in the obvious sense.
On $L\sp{p}$ estimates for integral transforms
T.
Walsh
195-215
Abstract: In a recent paper R. S. Strichartz has extended and simplified the proofs of a few well-known results about integral operators with positive kernels and singular integral operators. The present paper extends some of his results. An inequality of Kantorovič for integral operators with positive kernel is extended to kernels satisfying two mixed weak ${L^p}$ estimates. The ``method of rotation'' of Calderón and Zygmund is applied to singular integral operators with Banach space valued kernels. Another short proof of the fractional integration theorem in weighted norms is given. It is proved that certain sufficient conditions on the exponents of the $ {L^p}$ spaces and weight functions involved are necessary. It is shown that the integrability conditions on the kernel required for boundedness of singular integral operators in weighted ${L^p}$ spaces can be weakened. Some implications for integral operators in ${R^n}$ of Young's inequality for convolutions on the multiplicative group of positive real numbers are considered. Throughout special attention is given to restricted weak type estimates at the endpoints of the permissible intervals for the exponents.
Cubes with knotted holes
R. H.
Bing;
J. M.
Martin
217-231
Abstract: The statement that a knot $K$ has Property ${\rm {P}}$ means that (1) if $C$ is a cube with a
Endomorphism rings of projective modules
Roger
Ware
233-256
Abstract: The object of this paper is to study the relationship between certain projective modules and their endomorphism rings. Specifically, the basic problem is to describe the projective modules whose endomorphism rings are (von Neumann) regular, local semiperfect, or left perfect. Call a projective module regular if every cyclic submodule is a direct summand. Thus a ring is a regular module if it is a regular ring. It is shown that many other equivalent ``regularity'' conditions characterize regular modules. (For example, every homomorphic image is flat.) Every projective module over a regular ring is regular and a number of examples of regular modules over nonregular rings are given. A structure theorem is obtained: every regular module is isomorphic to a direct sum of principal left ideals. It is shown that the endomorphism ring of a finitely generated regular module is a regular ring. Conversely, over a commutative ring a projective module having a regular endomorphism ring is a regular module. Examples are produced to show that these results are the best possible in the sense that the hypotheses of finite generation and commutativity are needed. An application of these investigations is that a ring $R$ is semisimple with minimum condition if and only if the ring of infinite row matrices over $ R$ is a regular ring. Next projective modules having local, semiperfect and left perfect endomorphism rings are studied. It is shown that a projective module has a local endomorphism ring if and only if it is a cyclic module with a unique maximal ideal. More generally, a projective module has a semiperfect endomorphism ring if and only if it is a finite direct sum of modules each of which has a local endomorphism ring.