Department of Mathematics Server University of Manitoba

Research Profile

Dr. Thomas G. Kucera

Areas of Interest:

• Mathematical Logic: Model theory: Stability Theory (Classification Theory) and its applications: totally transcendental theories.
  Especially applications to abstract algebra
• Ring Theory: (one-sided) noetherian rings: injective modules over noetherian rings and non-commutative localization theory.
• Module Theory: modules over noetherian rings; injective modules over noetherian rings; pure-injective modules; totally transcendental modules.
• Model theory of modules: Structure of indecomposable tt modules. Elementary socles and radicals. Definable scalars. Elementary duality and dually tt theories.
• Infinitary methods in the model theory of modules and in abstract algebra in general:
  Infinitary elementary duality for modules (with Ph. Rothmaler).
  Saturated free algebras and almost indiscernible theories (with A. Pillay).

Details:

My main interests are in the model theory of modules and related subjects.

In general my research is focussed on using techniques from mathematical logic to help answer algebraic questions.

The main focus is primarily on totally transcendental modules. These are precisely the Sigma-pure-injective modules, and a general study of pure-injectives is also of interest. Model-theoretic properties of the category of modules over a ring can be described in terms of the Ziegler spectrum of the ring, that is, a topological space whose points are the isomorphism types of indecomposable pure-injective modules over the ring.

The most interesting examples of totally transcendental modules are the injective (right) modules over any (right) noetherian ring. It is already well known that the algebraic theory of these modules is quite difficult. I hope to be able to use techniques from the model theory of modules (in particular in relation to the manipulation of definable subsets) to study algebraic problems related to the structure and complexity of indecomposable injective modules over a (one-sided) noetherian ring. This is closely related to a famous unsolved problem: the status of Jacobson's conjecture for one-sided noetherian rings.

Current projects...

1. Infinitary definable properties of modules. With Ph. Rothmaler at CUNY, I am working on a detailed study of infinitary properties of modules: definable properties of infinite sequences; and properties of finite or infinite sequences definable by infinitary pp formulas. We explore elementary duality for such properties, and its relation to character duality.
2. Elementary socles and radicals. These are natural generalizations to a model-theoretic context of well-known and useful algebraic tools, introduced by Ivo Herzog. I have results connecting the internal structure of an indecomposable module with an elementary socle series, with the elementary radical structure of its character dual.
3. Modules over a QI-ring and Boyle's Conjecture. This is an old joint project with I. Herzog at Ohio State University, Lima. A QI ring is one over which every quasi-injective module is injective. It is not known whether there is a left QI ring which is not right QI. Boyle's Conjecture is that every left QI ring is left hereditary. We study the model theory of modules over a left QI ring, both out of a general interest in this class, and with the specific goal of resolving Boyle's conjecture.
4. Definable scalars. (definable scalar actions on a module, in particular on an (indecomposable) pure-injective modules. In some sense this is a form of localisation of the underlying ring. I am trying to find connections between the model-theoretically definable concepts and those commonly used in non-commutative algebra.
5. Modules with ACC on pp-formulas. These are the dually tt modules. Not much is known about them in general beyond what follows immediately by elementary duality.

Recent Graduate Students:

1. Alan Pasos Trejo, MSc, 2019.
2. Clint Enns, MSc, 2010.
   Thesis: Pure embeddings and pure-injectivity for topological modules.
3. Alina Duca, PhD, 2007.
   Thesis: Injective Modules over a Principal Left and Right Ideal Domain, with Applications.

2023–02–23