Einbettung lokal-beschränkter topologischer Ringe in Quotientenringe
Let R be a non-zero commutative ring, \({\mathcal T}^ a \)Hausdorff ring topology on R and H a multiplicatively closed subset of R consisting of regular elements, i.e. non-zero-divisors. If \({\mathcal T}\) can be extended to a ring topology of the quotient ring \(R_ H=H^{-1}\cdot R\) (in other words: if the topological ring (R,\({\mathcal T})\) can be embedded into \(R_ H)\), then all elements of H are topologically regular, i.e., they are not generalized zero-divisors. In general, this necessary condition for embeddability is not sufficient, as is shown by an example of \textit{V. I. Arnautov} [Mat. Issled. 48, 3-13 (1978; Zbl 0441.13015)]. But for locally bounded \({\mathcal T}\), we are able to construct an extension of \({\mathcal T}\) to \(R_ H\), if H satisfies the following stronger condition: H is a countable union of sets which are bounded and ''topologically equiregular''. A special case was given by Arnautov (loc. cit.), but the more general case can be applied to compact rings. For example, we obtain the following generalisation of a theorem of \textit{N. J. Rothman} [Math. Z. 91, 179-184 (1966; Zbl 0132.281)]: Every first-countable compact ring without zero-divisors can be embedded into its quotient field. Further, every compact noetherian ring can be embedded into its total quotient ring. The problem of embedding locally compact rings without zero- divisors into quotient fields turns out to reduce to the compact case: If there is a non-trivial solution, the ring must be compact. This theorem and its method of proof by means of topologically regular elements imply some theorems of \textit{Z. S. Lipkina} [Math. USSR, Izv. 1 (1967), 1187- 1208 (1969); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 31, 1239- 1262 (1967; Zbl 0192.383)] and \textit{S. Warner} [Trans. Am. Math. Soc. 139, 145-154 (1969; Zbl 0185.100), and J. Reine Angew. Math. 253, 146-151 (1972; Zbl 0232.13021)], namely that certain types of locally compact rings must also be compact or fields.
scientific article; zbMATH DE number 3877330
scientific article