Isomorphisms between infinite matrix rings (Q1063674)

In [J. Algebra 87, 261-281 (1984; Zbl 0554.16009)] the author obtained a categorical description of the isomorphisms between endomorphism rings of progenerators. As a special situation he described isomorphisms between matrix rings (over arbitrary rings) from a categorical point of view. In this paper the author succeeds in gaining a categorical description for the isomorphisms between endomorphism rings of countably infinitely generated free modules (over arbitrary rings), i.e. he proves the following main theorem: Let R, S be arbitrary associative rings with identity. Let \(U=R^{(N)}\), \(V=S^{(N)}\), where N is a countably infinite set. Let \(\Phi\) : End\({}_ R(U){\tilde \to}End_ S(V)\) be a ring isomorphism. Then there exists a unique (up to natural isomorphism) category equivalence \(F_{\Phi}: M_ R\to M_ S\) such that \(F_{\Phi}(U)=V\), and \(F_{\Phi}(f)=\Phi (f)\) for all \(f\in End_ R(U)\). - The author also shows that a semilinear description can be recovered from the categorical description in some particular cases.