We use the Drozd-Warfield structure theorem for finitely presented modules over a serial ring to investigate the model theory of modules over a serial ring, in particular, to give a simple description of pp-formulas and to classify the pure-injective indecomposable modules. We also study the question of whether every pure-injective indecomposable module over a valuation ring is the hull of a uniserial module.

Let Uq be the quantum group associated to sl2 with char≠2 and qk not a root of unity. The article is devoted to the model-theoretic study of the quantum plane kq[x,y], considered as an -structure, where is the language of representations of Uq. It is proved that the lattice of definable k-subspaces of kq[x,y] is complemented. This is deduced from the same result for the Uq-module M, which is defined to be the direct sum of all finite dimensional representations of (...) Uq. It follows that the ring of definable scalars for the quantum plane is a von Neumann regular epimorphic ring extension of the quantum group Uq. (shrink)

Let R be an associative ring with identity. It is shown that every Σ-cotorsion left R-module satisfies the descending chain condition on divisibility formulae. If R is countable, the descending chain condition on M implies that it must be Σ-cotorsion. It follows that, for countable R, the class of Σ-cotorsion modules is closed under elementary equivalence and pure submodules. The modules M that satisfy this descending chain condition are the cotorsion analogues of totally transcendental modules; we characterize them as the (...) modules M for which , for every countably presented flat module F. (shrink)

We characterize rings over which every cotorsion module is pure injective in terms of certain descending chain conditions and the Ziegler spectrum, which renders the classes of von Neumann regular rings and of pure semisimple rings as two possible extremes. As preparation, descriptions of pure projective and Mittag–Leffler preenvelopes with respect to so-called definable subcategories and of pure generation for such are derived, which may be of interest on their own. Infinitary axiomatizations lead to coherence results previously known for the (...) special case of flat modules. Along with pseudoflat modules we introduce quasiflat modules, which arise naturally in the model-theoretic and the category-theoretic contexts. (shrink)