Nilpotency and generalized Lie ideals
The authors generalize the notion of inner Lie ideal, defined by \textit{G. M. Benkart} [J. Algebra 43, 561-584 (1976; Zbl 0342.16009)], to that of generalized Lie ideal (GLI), and prove a structure theorem for such objects. For a nonempty subset V of a ring R, set \(V^{(1)}(R)=[V,R]\), and in general, \(V^{(n+1)}(R)=[V,V^{(n)}(R)]\). Call V a GLI of index n if V is an additive subgroup of R satisfying \(V^{(n)}(R)\subset V.\) The main result on the structure of GLIs is: Theorem. Let R be a prime ring with char R\(=0\), Z the center of R, C the extended centroid of R, and V a GLI of index at most n. Then either \(V\subset Z\), [I,R]\(\subset V\) for I a nonzero ideal of R, or \(V\subset C+S\) where \(S\subset RC+C\) and satisfies \(S^{3n-3}=0\). A necessary preliminary result of independent interest shows that when \(V^{(n)}(R)=0\), for V a subset, then the third condition of the theorem holds with the index of nilpotence of S equal to \([(n+1)/2]\).
scientific article; zbMATH DE number 4122166