Algebraic rings with compact topology
Let \(\Phi\) be a commutative associative ring with 1 (and \(1/2\) in the case of Jordan \(\Phi\)-algebras), which is also a Hausdorff topological ring. The author's main results are as follows: (1) Let \(\Gamma\) be a countable or compact ideal in \(\Phi\). If an alternative or Jordan compact topological \(\Phi\)-algebra is algebraic over \(\Gamma\), then it is algebraic over \(\Gamma\) of bounded index; and so such algebra is locally finite over \(\Gamma\). (2) In an alternative or Jordan compact ring, the McCrimmon radical coincides with the nil radical. Furthermore, every compact alternative or special Jordan nil-ring is solvable. Examples are given which show this last assertion is the best possible. Special cases of both these results were obtained earlier by \textit{M. I. Ursul} [Mat. Zametki 36, No. 6, 839--845 (1984; Zbl 0575.16022)].
scientific article; zbMATH DE number 3971004
scientific article