Some model theory over hereditary noetherian domains

Questions in the model theory of modules over hereditary noetherian domains are investigated with particular attention being paid to differential polynomial rings and to generalized Weyl algebras. We prove that there exists no isolated point in the Ziegler spectrum over a simple hereditary generalized Weyl algebra A of the sort considered by Bavula [Algebra i Analiz 4(1) (1992), 75–97] over a field k with char(k) = 0 (the first Weyl algebra A1(k) is one such) and the category of finite length modules over A does not have any almost split sequence. We show that the theory of all modules over a wide class of generalized Weyl algebras and related rings interprets the word problem for groups, and in the case that the field is countable there exists a superdecomposable pure-injective module over A. This class includes, for example, the universal enveloping algebra Usl2(k).