Philipp Rothmaler (CUNY), High and low formulas

Tue, 5 Oct 2021, 1 pm MDT

A partition of the set of unary positive primitive (pp) formulas for modules over an associative ring into four regions will be presented. These four types of formula have a bearing on various structural properties of modules, a few instances of which will be discussed in the talk. Domains, specifically Ore domains, turn out to play a prominent role.

One of the four types of formula are called high. These are used to define Ulm submodules and Ulm length of modules over an arbitrary associative ring. Pure injective modules turn out to have Ulm length at most 1 (just as in abelian groups). As a consequence, pure injective modules over RD domains (in particular, pure injective modules over the first Weyl algebra over a field of characteristic 0) decompose into a largest injective and a reduced submodule.