Prime ideals in a large class of nonassociative rings
Paul J.
Zwier
257-271
Abstract: In this paper a definition is given for a prime ideal in an arbitrary nonassociative ring $N$ under the single restriction that for a given positive integer $s \geqq 2$, if $A$ is an ideal in $N$, then ${A^s}$ is also an ideal. ($N$ is called an $s$-naring.) This definition is used in two ways. First it is used to define the prime radical of $ N$ and the usual theorems ensue. Second, under the assumption that the $ s$-naring $N$ has a certain property $ (\alpha )$, the Levitzki radical $L(N)$ of $N$ is defined and it is proved that $L(N)$ is the intersection of those prime ideals $P$ in $N$ whose quotient rings are Levitzki semisimple. $ N$ has property $ (\alpha )$ if and only if for each finitely generated subring $A$ and each positive integer $ m$, there is an integer $ f(m)$ such that ${A^{f(m)}} \subseteq {A_m}$. (Here ${A_1} = {A^s}$ and ${A_{ m + 1}} = A_m^s$.) Furthermore, conditions are given on the identities an $s$-naring $N$ satisfies which will insure that $N$ satisfies $(\alpha )$. It is then shown that alternative rings, Jordan rings, and standard rings satisfy these conditions.
Locally noetherian commutative rings
William
Heinzer;
Jack
Ohm
273-284
Abstract: This paper centers around the theorem that a commutative ring $ R$ is noetherian if every $ {R_P},P$ prime, is noetherian and every finitely generated ideal of $ R$ has only finitely many weak-Bourbaki associated primes. A more precise local version of this theorem is also given, and examples are presented to show that the assumption on the weak-Bourbaki primes cannot be deleted nor replaced by the assumption that Spec $(R)$ is noetherian. Moreover, an alternative statement of the theorem using a variation of the weak-Bourbaki associated primes is investigated. The proof of the theorem involves a knowledge of the behavior of associated primes of an ideal under quotient ring extension, and the paper concludes with some remarks on this behavior in the more general setting of flat ring extensions.
Some analytic varieties in the polydisc and the M\"untz-Szasz problem in several variables
Simon
Hellerstein
285-292
Abstract: For $1 \leqq {p_1} < {p_2} < \infty$ and $n \geqq 2$ it is shown that there exists a sequence of monomials $\{ \prod _{j = 1}^nS_j^\lambda mj\}$ with ${\lambda _{mj}} \sim m$ for each $j = 1, \ldots ,n$ whose linear span is dense in $ {L^{{p_1}}}({I^n})$ but not in $ {L^{{p_2}}}({I^n})$ ($ {I^n}$ is the Cartesian product of $n$ copies of the closed unit interval $[0, 1]$). Construction of the examples is via duality, making use of suitable analytic varieties in the polydisc.
Embedding a partially ordered ring in a division algebra
William H.
Reynolds
293-300
Abstract: D. K. Harrison has shown that if a ring with identity has a positive cone that is an infinite prime (a subsemiring that contains 1 and is maximal with respect to avoiding -- 1), and if the cone satisfies a certain archimedean condition for all elements of the ring, then there exists an order isomorphism of the ring into the real field. Our main result shows that if Harrison's archimedean condition is weakened so as to apply only to the elements of the cone and if a certain centrality relation is satisfied, then there exists an order isomorphism of the ring into a division algebra that is algebraic over a subfield of the real field. Also, Harrison's result and a related theorem of D. W. Dubois are extended to rings without identity; in so doing, it is shown that order isomorphic subrings of the real field are identical.
Necessary conditions for stability of diffeomorphisms
John
Franks
301-308
Abstract: S. Smale has recently given sufficient conditions for a diffeomorphism to be $\Omega$-stable and conjectured the converse of his theorem. The purpose of this paper is to give some limited results in the direction of that converse. We prove that an $\Omega$-stable diffeomorphism $ f$ has only hyperbolic periodic points and moreover that if $p$ is a periodic point of period $ k$ then the $k$th roots of the eigenvalues of $ df_p^k$ are bounded away from the unit circle. Other results concern the necessity of transversal intersection of stable and unstable manifolds for an $\Omega$-stable diffeomorphism.
Inductive definitions and computability
Thomas J.
Grilliot
309-317
Abstract: Sets inductively defined with respect to ${\prod _0},{\Sigma _1}$, (nonmonotonic) ${\prod _1}$ and $ {\Sigma _2}$ predicates are characterized in terms of the four chief notions of abstract recursion.
The space of retractions of the $2$-sphere and the annulus
Neal R.
Wagner
319-329
Abstract: Given a manifold $ M$, there is an embedding $ \Lambda$ of $ M$ into the space of retractions of $M$, taking each point to the retraction of $ M$ to that point. Considering $\Lambda$ as a map into the connected component containing its image, we prove that $\Lambda$ is a weak homotopy equivalence for two choices of $M$, namely, the $2$-sphere and the annulus.
Sufficiency classes of ${\rm LCA}$ groups
331-338
Abstract: By the sufficiency class $S(H)$ of a locally compact Abelian (LCA) group $ H$ we shall mean the class of LCA groups $G$ having sufficiently many continuous homomorphisms into $H$ to separate the points of $G$. In this paper we determine the sufficiency classes of a number of LCA groups and indicate how these determinations may help to describe the structure of certain classes of LCA groups. In particular, we give a new proof of a theorem of Robertson which states that an LCA group is torsion-free if and only if its dual contains a dense divisible subgroup. We shall also derive some facts about the compact connected Abelian groups and a result about topological $p$-groups containing dense divisible subgroups. We conclude by giving a necessary condition for two LCA groups to have the same sufficiency class.
Characteristic subgroups of lattice-ordered groups
Richard D.
Byrd;
Paul
Conrad;
Justin T.
Lloyd
339-371
Abstract: Characteristic subgroups of an $l$-group are those convex $l$-subgroups that are fixed by each $ l$-automorphism. Certain sublattices of the lattice of all convex $l$-subgroups determine characteristic subgroups which we call socles. Various socles of an $ l$-group are constructed and this construction leads to some structure theorems. The concept of a shifting subgroup is introduced and yields results relating the structure of an $ l$-group to that of the lattice of characteristic subgroups. Interesting results are obtained when the $l$-group is characteristically simple. We investigate the characteristic subgroups of the vector lattice of real-valued functions on a root system and determine those vector lattices in which every $l$-ideal is characteristic. The automorphism group of the vector lattice of all continuous real-valued functions (almost finite real-valued functions) on a topological space (a Stone space) is shown to be a splitting extension of the polar preserving automorphisms by the ring automorphisms. This result allows us to construct characteristically simple vector lattices. We show that self-injective vector lattices exist and that an archimedean self-injective vector lattice is characteristically simple. It is proven that each $l$-group can be embedded as an $l$-subgroup of an algebraically simple $ l$-group. In addition, we prove that each representable (abelian) $ l$-group can be embedded as an $l$-subgroup of a characteristically simple representable (abelian) $l$-group.
The existence of solutions of abstract partial difference polynomials.
Irving
Bentsen
373-397
Abstract: A partial difference (p.d.) ring is a commutative ring $ R$ together with a (finite) set of isomorphisms (called transforming operators) of $R$ into $R$ which commute under composition. It is shown here that (contrary to the ordinary theory [R. M. Cohn, Difference algebra]) there exist nontrivial algebraically irreducible abstract p.d. polynomials having no solution and p.d. fields having no algebraically closed p.d. overfield. If $F$ is a p.d. field with two transforming operators, then the existence of a p.d. overfield of $F$ whose underlying field is an algebraic closure of that of $F$ is a necessary and sufficient condition for every nontrivial algebraically irreducible abstract p.d. polynomial $P$ in the p.d. polynomial ring $ F\{ {y^{(1)}},{y^{(2)}}, \ldots ,{y^{(n)}}\}$ to have a solution $\eta$ (in some p.d. overfield of $ F$) such that: $ \eta$ has $n - 1$ transformal parameters, $ \eta$ is not a proper specialization over $F$ of any other solution of $P$, and, if $Q$ is a p.d. polynomial whose indeterminates appear effectively in $P$ and $Q$ is annulled by $\eta$, then $Q$ is a multiple of $P.P$ has at most finitely many isomorphically distinct such solutions. Necessity holds if $ F$ has finitely many transforming operators.
Word problem for ringoids of numerical functions
A.
Iskander
399-408
Abstract: A. The composition ringoid of functions on (i) the positive integers, (ii) all integers, (iii) the reals and (iv) the complex numbers, do not satisfy any identities other than those satisfied by all composition ringoids. B. Given two words $u,\upsilon$ of the free ringoid, specific functions on the positive integers, ${f_1}, \ldots ,{f_k}$ can be described such that $u({f_1}, \ldots ,{f_k})$ and $\upsilon ({f_1}, \ldots ,{f_k})$, evaluated at 1, are equal iff $u = \upsilon$ is an identity of the free ringoid.
An approach to the polygonal knot problem using projections and isotopies
L. B.
Treybig
409-421
Abstract: The author extends earlier work of Tait, Gauss, Nagy, and Penney in defining and developing properties of (1) the boundary collection of a knot function, and (2) simple sequences of knot functions or boundary collections. The main results are (1) if two knot functions have isomorphic boundary collections then the knots they determine are equivalent, and (2) if two knot functions determine equivalent knots, then the given functions (their boundary collections) are the ends of a simple sequence of knot functions (boundary collections). Matrices are also defined for knot functions.
Concerning a bound problem in knot theory
L. B.
Treybig
423-436
Abstract: In a recent paper Treybig shows that if two knot functions $ f,g$ determine equivalent knots, then $f,g$ are the ends of a simple sequence $ x$ of knot functions. In an effort to bound the length of $x$ in terms of $f$ and $g$ (1) a bound is found for the moves necessary in moving one polyhedral disk onto another in the interior of a tetrahedron and (2) it is shown that two polygonal knots $K,L$ in regular position can ``essentially'' be embedded as part of the $1$-skeleton of a triangulation $T$ of a tetrahedron, where (1) all 3 cells which are unions of elements of $T$ can be shelled and (2) the number of elements in $T$ is determined by $K,L$. A number of ``counting'' lemmas are proved.
Bounded holomorphic functions of several complex variables. I
Dong Sie
Kim
437-443
Abstract: A domain of bounded holomorphy in a complex analytic manifold is a maximal domain for which every bounded holomorphic function has a bounded analytic continuation. In this paper, we show that this is a local property: if, for each boundary point of a domain, there exists a bounded holomorphic function which cannot be continued to any neighborhood of the point, then there exists a single bounded holomorphic function which cannot be continued to any neighborhood of the boundary points.
The product theorem for topological entropy
L. Wayne
Goodwyn
445-452
Whitehead products as images of Pontrjagin products
Martin
Arkowitz
453-463
Abstract: A method is given for computing higher order Whitehead products in the homotopy groups of a space $X$. If $X$ can be embedded in an $H$-space $E$ such that the pair $(E,X)$ has sufficiently high connectivity, then we prove that a higher order Whitehead product element in the homotopy of $X$ is the homomorphic image of a Pontrjagin product in the homology of $E$. The two main applications determine a higher order Whitehead product element in (1) ${\pi _ \ast }(B{U_t})$, the homotopy groups of the classifying space of the unitary group ${U_t}$, (2) the homotopy groups of a space with two nonvanishing homotopy groups.
Engulfing continua in an $n$-cell
Richard J.
Tondra
465-479
Abstract: In this paper it is shown that there exist open connected subsets ${D_1},{D_2}$, and ${D_3}$ of an $n$-cell $E$ such that, if $C$ is any proper compact connected subset of $ E$ and $C \subset U,U$ open, then there exists a homeomorphism $ h$ of $E$ onto itself such that $C \subset h({D_i}) \subset U$ for some $i,1 \leqq i \leqq 3$.
Quasiconformal mappings and Royden algebras in space
Lawrence G.
Lewis
481-492
Abstract: On every open connected set $G$ in Euclidean $n$-space ${R^n}$ and for every index $p > 1$, we define the Royden $ p$-algebra ${M_p}(G)$. We use results by F. W. Gehring and W. P. Ziemer to prove that two such sets $ G$ and $G'$ are quasiconformally equivalent if and only if their Royden $n$-algebras are isomorphic as Banach algebras. Moreover, every such algebra isomorphism is given by composition with a quasiconformal homeomorphism between $G$ and $G'$. This generalizes a theorem by M. Nakai concerning Riemann surfaces. In case $p \ne n$, the only homeomorphisms which induce an isomorphism of the $p$-algebras are the locally bi-Lipschitz mappings, and for $1 < p < n$, every such isomorphism arises this way. Under certain restrictions on the domains, these results extend to the Sobolev space $H_p^1(G)$ and characterize those homeomorphisms which preserve the $H_p^1$ classes.
Differentiable monotone maps on manifolds. II
P. T.
Church
493-501
Abstract: Let ${M^n}$ and ${N^n}$ be closed manifolds, and let $ G$ be any (nonzero) module. (1) If $ f:{M^3} \to {N^3}$ is $ {C^3}$ $G$-acyclic, then there is a closed $ {C^3}$ $3$-manifold ${K^3}$ such that $ {N^3}\char93 {K^3}$ is diffeomorphic to ${M^3}$, and $ {f^{ - 1}}(y)$ is cellular for all but at most $r$ points $ y \in {N^3}$, where $ r$ is the number of nontrivial $G$-cohomology $3$-spheres in the prime decomposition of $ {K^3}$. (2) If $f:{M^3} \to {M^3}$ or $f:{S^3} \to {M^3}$ is $G$-acyclic, then $f$ is cellular. In case $G$ is $Z$ or ${Z_p}$ ($p$ prime), results analogous to (1) and (2) in the topological category have been proved by Alden Wright. (3) If $ f:{M^n} \to {M^n}$ or $f:{S^n} \to {M^n}$ is real analytic monotone onto, then $f$ is a homeomorphism.
Invariant states
Richard H.
Herman
503-512
Abstract: States of a ${C^ \ast }$-algebra invariant under the action of a group of automorphisms of the ${C^ \ast }$-algebra are considered. It is shown that ``clustering'' states in the same part are equal and thus the same is true of extremal invariant states under suitable conditions. The central decomposition of an invariant state is considered and it is shown that the central measure is mixing if and only if the state satisfies a strong notion of clustering. Under transitivity of the central measure and some reasonable restrictions, the central decomposition is the ergodic decomposition of the state with respect to the isotropy subgroup.