Residue class domains of the ring of convergent sequences and of \(C^{\infty}([0,1],{\mathbb{R}})\)
Let \(N^*\) denote the one-point compactification of the integers N, and \(C(N^*,R)\) denote the ring of continuous mappings from \(N^*\) into the real numbers R, that is to say, \(C(N^*,R)\) is the ring of real convergent sequences. The author studies the residue class domains of \(C(N^*,R)\) and shows that there are precisely ten up to isomorphism. In this work, significant roles are played by non-principal ultrafilters on N and the Ax-Kochen theory of valuation fields. The author uses the following observation to extend these results: if X is a non-discrete metric space, then \(N^*\) can be C-embedded in X, and every residue class domain of \(C(N^*,R)\) is a residue class domain of C(X,R). Furthermore, if X is a topological space with a compact non-discrete subset, then C(X,R) includes four of the residue class rings of \(C(N^*,R)\). More results of this kind are proved for other classes of spaces X. The author then turns to the ring \(C^{\infty}(X,R)\) of \(C^{\infty}\)-functions on a closed interval X. Using rather intricate constructions, he classifies up to isomorphism twenty residue class domains of \(C^{\infty}(X,R)\), which are derived from those of \(C(N^*,R)\), and shows by means of an example that there are still more of them.
scientific article; zbMATH DE number 4209057