Modules and representations of nonassociative compact topological rings (Q1091459)

The necessary definitions can be found in the monograph of \textit{K. A. Zhevlakov}, \textit{A. M. Slin'ko}, \textit{I. P. Shestakov}, and \textit{A. I. Shirshov}, Nearly associative rings (Moscow, Nauka, 1978; Zbl 0445.17001); English translation Academic Press (1982; Zbl 0487.17001), chapter 11. It is proved that under certain restrictions on a variety \({\mathcal M}\) of rings every topologically irreducible locally compact R-module, R being a compact ring from \({\mathcal M}\), is finite. Using this result the following characterization of the quasiregular radical J(R) of a compact ring R from \({\mathcal M}\) is given, where \({\mathcal M}\) is the variety of alternative or Jordan rings: J(R) coincides with the intersection of all locally compact irreducible representations of the ring R in the variety \({\mathcal M}\).