We consider separably closed fields of characteristic $p > 0$ and fixed imperfection degree as modules over a skew polynomial ring. We axiomatize the corresponding theory and we show that it is complete and that it admits quantifier elimination in the usual module language augmented with additive functions which are the analog of the $p$-component functions.

In Dellunde et al. 997–1015), we determined the complete theory Te of modules of separably closed fields of characteristic p and imperfection degree e, eω{∞}. Here, for 0≠eω, we describe the closed set of the Ziegler spectrum corresponding to Te. Further, we establish a correspondence between certain submodules and n-types and we investigate several notions of dimensions and their relationships with the Lascar rank. Finally, we show that Te has uniform p.p. elimination of imaginaries and deduce uniform weak elimination of (...) imaginaries. (shrink)

Rings which, from the ring-theoretic point of view, are very different may well have categories of modules which are extremely similar. More generally, the category of modules over a ring may contain many other categories of modules. Ideas from model theory are of use in elucidating this state of affairs. In particular we investigate the model-theoretic effect of tilting functors between categories of modules.

Elementary duality between left and right modules over a ring, especially its interpretation in terms of the relevant functor categories, is discussed, as is the relationship between these categories of functors and sorts in theories of modules. A topology on the set of indecomposable pure-injective modules over a ring is introduced. This topology is dual to the Ziegler topology and may be seen as a generalisation of the Zariski topology.