October 18
Philipp Rothmaler,
CUNY
High and low formulas in the model theory of modules
A positive primitive (henceforth pp) formula is an existentially quantified (i.e., a projection of a) finite system of linear equations (over a given associative ring R). In this talk I am interested exclusively in such formulas with one free variable. I call such a formula high if, in every injective module E, it defines all of E. Note, the high formulas form a filter in the lattice of all unary pp formulas. Long time ago I discovered this dichotomy: every (unary) pp formula is either high or else bounded (but not both), which means that there is a nonzero ring element that annihilates, in every module, all of the set defined by the formula (which is, in fact, an additive subgroup of the module).
I had not given this much further thought until recently, when I discovered, in collaboration with A. Martsinkovsky, that the dual notion of low formula gives rise to a torsion theory, namely injective torsion as introduce by him and J. Russell in recent work. I call a formula low if it vanishes on the flat modules, or, equivalently, on the ring as a module over itself. Note, the low formulas form an ideal in the lattice of all unary pp formulas. I will explain how elementary duality (as introduced by Prest and Herzog) yields at once another dichotomy: every pp formula is either low or else cobounded (but not both), where this means, dually, that the action of some nonzero ring element sends every module into the subgroup defined by that formula.
Interestingly, these two dichotomies are, in general, completely independent. But I will show how their interplay can be used to characterize (not necessarily commutative) domains within the class of all rings, and one-sided Ore domains and also two-sided Ore domains within all domains. (Commutative domains are two-sided Ore.)