Dissertation Committee: Thomas Scanlon (Chair), Leo Harrington, Christos Papadimitriou
We concern ourselves with algebraic structures which possess two strong model-theoretic properties: stability and aleph_{0}-categoricity (countable categoricity). In the case of rings (not assumed to be commutative, nor having an identity), we provide a new proof of Baldwin and Rose's result that a stable, aleph_{0}-categorical ring is nilpotent-by-finite. In the case of groups, Baur, Cherlin, and Macintyre and, separately, Felgner, showed that a group with these two properties is nilpotent-by-finite. The first three authors conjectured further that such a group is abelian-by-finite. We prove that this conjecture is equivalent to the corresponding ring conjecture, namely that such a ring is null-by-finite (i.e. trivial multiplication up to a finite ring extension). We then describe structural restrictions on group counterexamples to the BCM-conjecture. In the process, we develop the theory of quasiendomorphism rings, describe the multiple situations in which they arise in these counterexamples, and use them to translate questions about the structure of counterexamples into questions about simultaneous solutions to systems of linear equations over finite fields.
[collapse]