The goal of this work was to prove the fact, that definable closure of any subclass of cotorsion modules closed under direct sums consists of $\Sigma$-cotorsion modules. The only known proof uses substantially the calculus of derived category, in this work we tried to prove the same, but only by means of a given category of all right $R$-modules and set-theoretic properties of partial orders indexing direct systems of $R$-modules. The main results of this work are proved under additional assumptions on the ring $R$, in particular $\vert R\vert\leq\aleph_{\omega}$ or $\text{dim}(R)<\aleph_{\omega}$. Attempts to give s proof in the same general situation, where the fact is known to hold, was not successful. Powered by TCPDF (www.tcpdf.org)

Any set of finitely presented left modules defines a relative purity for left modules and also apurity for right modules. Purities defined by various classes are compared and investigated,especially in the contexts of modules over semiperfect rings and over tame hereditary, andmore general, finite-dimensional algebras. Connections between the indecomposable relativelypure-injective modules and closure in the full support topology (a refinement of theZiegler spectrum) are described.Duality between left and right modules is used to define the concept of a class of leftmodules and a class of right modules forming an almost dual pair. Definability of suchclasses is investigated, especially in the case that one class is the closure of a set of finitelypresented modules under direct limits. Elementary duality plays an important role here.Given a set of finitely presented modules, the corresponding proper class of relativelypure-exact sequences can be used to define a relative notion of cotorsion pair, which weinvestigate.The results of this thesis unify and extend a wide range of results in the literature.