Thomas S. Shores

Elementary divisor rings and finitely presented modules

Throughout, rings are commutative with unit and modules are unital. We prove that R is an elementary divisor ring if and only if every finitely presented module over R is a direct sum of cyclic modules, thus providing a converse to a theorem of Kaplansky and answering a question of Warfield. We show that every Bezout ring with a finite number of minimal prime ideals is Hermite. So, in particular, semilocal Bezout rings are Hermite answering affirmatively a question of Henriksen. We show that every semihereditary Bezout ring is Hermite. Semilocal adequate rings are characterized and a partial converse to a theorem of Henriksen is established.

Rings whose finitely generated modules are direct sums of cyclics

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

A new sinc-galerkin method for convection-diffusion equations with mixed boundary conditions

Abstract A new sinc-Galerkin method is developed for approximating the solution of convection-diffusion equations with mixed boundary conditions on half-infinite intervals. The method avoids differentiation of the coefficients of the PDE, rendering it appropriate as a forward solver in an inverse coefficient problem. The method has the advantage that no functions are appended to the sinc basis in the discretization. An error analysis is included, and it is shown that the error in the approximate solution is bounded in the infinity norm by the norm of the inverse of the coefficient matrix multiplied by a factor that decays exponentially with the size of the system. We demonstrate the exponential convergence of the method on several test problems.

Arithmetical and semihereditary semigroup rings

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.

Numerical methods for an ion transport problem

Abstract We consider the numerical solution to the nonlinear differential equation v′′+ 1 2 ξv′+ 1 2 ∫ 0 ∞ v(ξ) d ξh(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) d ξ.

Decompositions of finitely generated modules

A commutative ring with unit is called a d-ring if every finitely generated Loewy module is a direct sum of cyclic submodules. It is shown that every d-ring is a T-ring, i.e., Loewy modules over such rings satisfy a primary decomposition theorem. Some applications of this result are given.

Numerical study of reaction-mineralogy-porosity changes in porous media

We develop a one-dimensional, nonlinear, nonlocal model that governs the advection and dispersion of a chemical solute that reacts with a porous, mineral fabric to produce a change in porosity and a change in mineralogy. The flow is either driven by a pressure gradient, consistent with Darcys law, or an imposed boundary velocity. Constitutive equations based on geometric considerations relate volume changes to the replacement reactions that occur. Numerical methods are used to analyze the behavior of the system.

Uniqueness and numerical recovery of a potential on the real line

We consider the recovery of the potential q(x) in the singular problem Under suitable conditions the coefficient is uniquely determined from the set of flux data at the origin corresponding to the source terms , which constitute a basis for . To facilitate numerical recovery, the problem is formulated as an infinite-dimensional least-squares minimization problem. Tikhonov regularization is employed, and the resulting problem is discretized via sinc collocation. Several numerical examples are included.

Decompositions of modules and matrices

A canonical form for a module M over a commutative ring R is a decomposition M £ R/^

An Inverse Coefficient Problem for an Integro-differential Equation

In this article we consider the inverse coefficient problem of recovering the function { ( x ) system of partial differential equations that can be reduced to a second order integro-differential equation