Elementary groups and stability for Jordan pairs
For \(n\geq \max\{ 2,\text{stable rank}\}\), the ideal structure of an associative ring \(R\) is known to have direct impact on the normal subgroup structure of the elementary group \(E_n (R)\) [see \textit{A. J. Hahn} and \textit{O. T. O'Meara}, The classical groups and \(K\)-theory (Springer Grundlehren Math. Wiss. 291) (1989; Zbl 0683.20033)]. An associative ring \(R\) is said to be of stable rank 1 in case for any \(r_1, r_2\in R\) such that there are \(t_1, t_2\in R\) with \(t_1 r_1+ t_2 r_2=1\), there is an element \(t\in R\) with \(R(r_1+ tr_2) =R\). This notion was extended to alternative rings by \textit{J. Faulkner} [Geom. Dedicata 14, 177-188 (1983; Zbl 0519.17012)]. In the paper under review, this is further extended to Jordan algebras and pairs. To do so, a so called projective elementary group is attached to any Jordan pair, slightly changing the definition given by Faulkner in the reference above, and which is the substitute for the projectivized group \(E_2 (R)\) for associative \(R\). The notion of stability thus defined, which is thoroughly studied, is such that a unital associative or alternative algebra \(R\) has stable rank 1 if and only if the associated Jordan pair \((R,R)\) is stable. As the author remarks: ``There appears to be no general way to associate groups with Jordan pairs which would correspond to higher order matrix groups for rings. For this reason, it seem hopeless to define higher stable rank conditions for Jordan pairs''. The author also gives a useful characterization of stability in terms of a notion of quasiinvertibility for quadruples of elements and relates stability and Peirce decompositions.
quasiinvertibility for quadruples of elements
scientific article; zbMATH DE number 784415
scientific article