Herstein's Lie theory revisited
This well written paper considers the Lie structure of the skew-symmetric elements K in a semi-prime ring R with involution, *, and satisfying \(2R=R\). The purpose of the paper is to redo this structure theory by using a somewhat different approach from that which has developed from the initial work in this area by \textit{I. N. Herstein} [Am. J. Math. 78, 629-649 (1956; Zbl 0071.259)]. The approach of the authors requires the theory of generalized polynomial identities (GPIs) for rings with involution and yields a uniform and coherent treatment which avoids most of the complex and detailed computations of the existing literature on the subject. One knows that Lie subideals U of K contain Lie ideals of the form [I\(\cap K,K]\), for I a *-invariant ideal of R depending on U. The basic difficulty is to show that R is quite special if [I\(\cap K,K]\) is Lie solvable. The authors give a nice formulation of these results. In showing that U contains [I\(\cap K,K]\), for \(U\triangleleft V\triangleleft K\), the authors construct I in essentially the standard way, but their description is quite clear, and introduces a choice depending on the type of involution on R. When [I\(\cap K,K]\subset Z(K)\), the center of K, and when R is *-prime, the theory of GPIs is used to obtain a precise description of the possibilities for the (algebraically closed) *-central *-closure of R; this is essentially the classical situation of matrices over fields. In obtaining this description, the authors use results of \textit{W. E. Baxter} and \textit{W. S. Martindale III} [Commun. Algebra 10, 847-874 (1982; Zbl 0482.16014)] to handle the case of involutions of the second kind, rather than the usual approach of using the Lie structure of R. Even when [I\(\cap K,K]\subset Z(K)\), the authors show that one can still construct a Lie ideal of K in U. Using this they obtain their main theorem which states that K/Z(K) is a semi-prime Lie ring in which every nonzero n-subideal \((U_ n\triangleleft...\triangleleft U_ 1\triangleleft K/Z(K))\) contains a nonzero ideal.
scientific article; zbMATH DE number 3922906