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Papers by Thomas Shores

Archiv der Mathematik, Dec 1, 1972

Rocky Mountain Journal of Mathematics, Mar 1, 1973

In this article a chain condition for groups, called the bounded chain condition, is studied; thi... more In this article a chain condition for groups, called the bounded chain condition, is studied; this chain condition includes the ascending and descending chain conditions as special cases. It is shown that every locally radical group which satisfies the bounded chain condition on subgroups must satisfy the ascending or descending chain condition on subgroups, i.e., such groups are Artinian or Noetherian. The same conclusion holds for nilpotent groups which satisfy the bounded chain condition on abelian subgroups.

Canadian Journal of Mathematics, Dec 1, 1980

Throughout this paper the ring R and the semigroup S are commutative with identity; moreover, it ... more Throughout this paper the ring R and the semigroup S are commutative with identity; moreover, it is assumed that S is cancellative, i.e., that S can be embedded in a group. The aim of this note is to determine necessary and sufficient conditions on R and 5 that the semigroup ring R[S] should be one of the following types of rings: principal ideal ring (PIR), ZPI-ring, Bezout, semihereditary or arithmetical. These results shed some light on the structure of semigroup rings and provide a source of examples of the rings listed above. They also play a key role in the determination of all commutative reduced arithmetical semigroup rings (without the cancellative hypothesis on S) which will appear in a forthcoming paper by Leo Chouinard and the authors [4]. Our results are motivated in large part by the paper [11] of R. Gilmer and T. Parker. In particular, Theorem 1.1 of [11] asserts that if R and S are as above and, moreover, if 5 is torsion-free, then the following are equivalent conditions: (1) R[S] is a Bezout ring; (2) R[S] is a Priifer ring; (3) R is a (von Neumann) regular ring and 5 is isomorphic to either a subgroup of the additive rationals or the positive cone of such a subgroup. One could very naturally include a fourth condition, namely: (4) R[S] is arithmetical. L. Fuchs [7] defines an arithmetical ring as a commutative ring with identity for which the ideals form a distributive lattice. Since a Priifer ring is one for which (A + B) C\ C = {A C\ C) + (B Pi C) whenever at least one of the ideals A, B or C contains a regular element (see [18]), arithmetical rings are certainly Priifer. On the other hand, it is well known that every Bezout ring is arithmetical, so that (4) is indeed equivalent to (l)-(3) in Theorem 1.1. In Theorem 3.6 of this paper we drop the requirement that S be torsion-free and determine necessary and sufficient conditions for the semigroup ring of a cancellative semigroup to be arithmetical. Examples are included to show that for these more general semigroup rings, the equivalences of the torsion-free case are no longer true. Theorems 4.1 and 4.2 provide characterizations of semigroup rings that are ZPI-rings and PIR's. Again, the corresponding results in [18] for torsion-free semigroups fail to hold in the more general case. We would like to thank Leo Chouinard for showing us how to remove

Acta Mathematica Hungarica, Sep 1, 1976

Proceedings of the American Mathematical Society, Mar 1, 1971

A commutative ring with unit is called a d-r'mg if every finitely generated Loewy module is a dir... more A commutative ring with unit is called a d-r'mg if every finitely generated Loewy module is a direct sum of cyclic submodules. It is shown that every á-ring is a T-ring, i.e., Loewy modules over such rings satisfy a primary decomposition theorem. Some applications of this result are given.

Michigan Mathematical Journal, Jul 1, 1975

Applied Numerical Mathematics, Feb 1, 2000

We consider the numerical solution to the nonlinear differential equation v′′+12ξv′+12∫0∞v(ξ)dξh(... more We consider the numerical solution to the nonlinear differential equation v′′+12ξv′+12∫0∞v(ξ)dξh(v)′=0, with boundary conditions v(0)=1 and v(∞)=0. This problem arises from a model for ion transport. An iterative scheme is developed for this problem and convergence is proved. Finally, Sinc methods are used to implement the numerical scheme for this problem and computation of the current response term ∫0∞v(ξ)dξ.

Communications in Algebra, 1980

In this paper conditions on the commutative ring R (with identity) and the commutative semigroup ... more In this paper conditions on the commutative ring R (with identity) and the commutative semigroup ring S (with identity) are found which characterize those semigroup rings R[S] which are reduced or have weak global dimension at most one. Likewise, those semigroup rings R[S] which are semihereditary are completely determined in terms of R and S.

Duke Mathematical Journal, Dec 1, 1974

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Communications in Algebra, 1981

Mathematical Models and Methods in Applied Sciences, Feb 1, 1993

We study the existence of travelling waves that are produced by moving sources in a nonlinear rea... more We study the existence of travelling waves that are produced by moving sources in a nonlinear reactive-convective model of reactive flow. The analysis leads to a non-autonomous differential-algebraic system and the existence of solutions is proved using a topological theorem on terminal boundary value problems on ℝ. We also consider an inverse-problem and determine a moving source that leads to a given travelling wave in the system. The ad hoc methods developed herein to analyze the nonautonomous system may be applicable to other problems on infinite domains.

Australian & New Zealand industrial and applied mathematics journal, Jul 18, 2004

We consider the identification of scattering and absorption rates in the stationary radiative tra... more We consider the identification of scattering and absorption rates in the stationary radiative transfer equation. For a stable solution of this parameter identification problem, we consider Tikhonov regularization within Banach spaces. A regularized solution is then defined via an optimal control problem constrained by an integro partial differential equation. By establishing the weak-continuity of the parameter-to-solution map, we are able to ensure the existence of minimizers and thus the well-posedness of the regularization method. In addition, we prove certain differentiability properties, which allow us to construct numerical algorithms for finding the minimizers and to analyze their convergence. Numerical results are presented to support the theoretical findings and illustrate the necessity of the assumptions made in the analysis.

Applicable Analysis, Feb 1, 1993

ABSTRACT This paper examines the existence of steady state solutions in the Fickett-Majda model o... more ABSTRACT This paper examines the existence of steady state solutions in the Fickett-Majda model of detonation-combustion when heat sources are present and the chemical kinetics is governed by a single, reversible reaction. The problem reduces to studying the bifurcations of a nonlinear algebraic equation, and conditions for existence are obtained in terms of the heat release parameter and total energy of the source

Canadian mathematical bulletin, Feb 1, 1969

Bulletin of the American Mathematical Society, Nov 1, 1973

Springer eBooks, 2018

All rights reserved. This work may not be translated or copied in whole or in part without the wr... more All rights reserved. This work may not be translated or copied in whole or in part without the written permission for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except proprietary identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to rights.

University Microfilms, Inc. eBooks, 1968

ABSTRACT Thesis (Ph. D.)--University of Kansas, Mathematics, 1968. Includes bibliographical refer... more ABSTRACT Thesis (Ph. D.)--University of Kansas, Mathematics, 1968. Includes bibliographical references.

Pacific Journal of Mathematics, Mar 1, 1972

Let R be a ring with unit whose right and left ideals are two-sided ideals. It is shown that ever... more Let R be a ring with unit whose right and left ideals are two-sided ideals. It is shown that every Noetherian injective ϋNmodule has finite length (i.e., has a finite composition series). If I is a maximal ideal of R, then R has a universal localization, Ri at I. The condition that the injective hull of R/I is finite is characterized in terms of Ri. l Introduction* In this note all rings have unit and modules are unital right modules. A. Rosenberg and D Zelinsky have shown that if R is a commutative ring and / is a maximal ideal of R, then the injective hull of R/I is finite (i.e., has finite length) if and only if the localization of R at I is Artinian (see Theorem 5 of [6, p. 379]). In this note we shall prove an extended version (Theorem 4) of their result for a class of rings which is somewhat interesting in itself. Let us call a ring R a duo ring if xR-Rx for all x e R (equivalently all ideals are bilateral). Such rings were investigated by E. Feller [2] and G. Thierrin [7] Trivial examples of duo rings are, of course, commutative rings and division rings. Nontrivial duo rings are not difficult to come by (e.g., any noncommutative special primary ring is duo, since the only right or left ideals are powers of the unique maximal ideal). In fact some interesting examples of duo rings have already occurred in the literature: M. Auslander and 0. Goldman have shown in [1, p. 13] that there exist noncommutative maximal orders which are both duo rings and Noetherian domains. Further investigations of such rings have been carried out by G. Maury in [4]. One of the basic difficulties in extending Rosenberg and Zelinsky's result to duo rings is the existence of suitable localizations. This problem is considered in §2. Next we show in §3 that Noetherian duo rings are classical in the sense that the familiar primary decomposition theory of commutative Noetherian rings extends to duo rings. We use this fact to show that Noetherian injective modules over duo rings are finite. Finally we prove our main result in §4. The injective hull of the module M will be denoted by E(M). If A and B are subsets of M or R, then A .' B = {xe R\xB S A) and A\B-{xeR\Ax QB). Also A\B is the set of elements in A but not B. 2* Localizations* First of all we need a suitable definition of the term "localization." The ideal P of R is prime if AB g P implies igPorίgPfor all ideals A and B of R.

Proceedings of the American Mathematical Society, 1974

It is shown that every commutative semihereditary Bezout ring of Krull dimension at most one is a... more It is shown that every commutative semihereditary Bezout ring of Krull dimension at most one is an elementary divisor ring. A consequence is that the ring of polynomials in one indeterminate over a von Neumann regular ring is an elementary divisor ring. All rings of this note are commutative with unity and modules are unital. Recently there has been some interest in the polynomial ring R[X\ where R

Journal of Algebra, Oct 1, 1974

All rings in this discussion are commutative with unity, and all modules are unital. The purpose ... more All rings in this discussion are commutative with unity, and all modules are unital. The purpose of this paper is to study the structure of those rings whose modules satisfy the conclusion of a classical theorem well known to every beginning student of algebra, namely: Every finitely generated module over a principal ideal domain is a direct sum of cyclic submodules. Let us canonize this theorem with the following terminology: A ring is said to be an FGC-ring (or to have FGC) if every finitely generated module over the ring is a direct sum of cyclic submodules. What can be said about such rings ? In particular, how far are they from being principal ideal rings ? As a point of departure, we review the known results on FGC-rings. (The reader is referred to Section 1 for the relevant definitions.) A major step beyond the classical theorem was taken by Kaplansky, who proved in [l] and [2] that a local domain has FGC if and only if it is an almost maximal valuation ring. Subsequently, Matlis [3] generalized this theorem by showing that an h-local domain has FGC if and only if it is Bezout and every localization is almost maximal. An example of such a ring, neither a local ring nor a principal ideal ring, was provided by Osofsky [4]. Recently Gill [5] and Lafon [6] completely disposed of the local problem by generalizing Kaplansky's theorem to arbitrary local rings. In another direction, Pierce [7] has characterized the von Neumann regular FGC-rings as finite direct products of fields. It is interesting to observe (see Section 1 for details) that all known examples of FGC-rings satisfy a module-theoretic version of another classical theorem, namely the elementary divisor theorem for matrices over a principal ideal domain. To be precise, we define a canonical form for an R-module ilP to be a decomposition ME R/I, @ ... @ R/In , where I1 _C I, C ... _C I, f R. A ring R is then called a CF-ring (or said to have CF) if every direct sum of * The Author's research was partially supported by an NSF grant.