Constructing division rings as module-theoretic direct limits (Q2781399)

A construction of division rings as quotients of localizations was given in [\textit{P. M. Cohn}, Free rings and their relations, Academic Press (1971; Zbl 0232.16003); see also \textit{P. Malcolmson}, J. Algebra 64, 399-413 (1980; Zbl 0442.16015)]. The author's aim is to accomplish such a construction by means of closure operators. Let \(R\) be any ring and \(f\colon R\to D\) a homomorphism to a division ring. The author defines a closure operator on free \(R\)-modules; this closure operator is shown to satisfy the usual properties of algebraic closure (with exchange property), and it leads to the notion of a coherent matroidal structure on free \(R\)-modules. For a given homomorphism \(f\colon R\to D\) (to a division ring) the author describes a subclass of pointed \(R\)-modules, from which \(D\) can be reconstructed as the endomorphism ring of their direct limit. Conversely, he shows that from a coherent matroidal structure on the free \(R\)-modules this construction can be repeated to produce a division ring \(D\) with a map from \(R\) to \(D\), and that in fact these two constructions are inverse to each other. These ideas are illustrated by constructions of division rings from Ore domains and firs (free ideal rings). In the latter case there may be many non-isomorphic division rings of fractions, related by specializations, and the relation of their matroidal structures is examined. This is an important construction which requires careful study for its full appreciation. The main result was found by the author in 1977.