On invariant additive subgroups
The author obtains the most comprehensive structure theorem to date for additive subgroups of prime rings which are invariant under certain kinds of automorphisms. Specifically, let A be an additive subgroup of the prime ring R and assume that for every \(x\in R\) satisfying \(x^ 2=0\), \((1+x)A(1-x)\subset A\). If Q is the Martindale quotient ring of R, let \(N=\{q\in Q|\) \(Iq+qI\subset R\) for some nonzero ideal I of \(R\}\). The main theorem asserts that when N contains a nontrivial idempotent then either A is central, A contains a noncommutative Lie ideal of R, or char R\(=2\) and R satisfies the standard identity \(S_ 4\). The author proceeds to analyze the last possibility and determines a condition on the center of R which is equivalent to the statement that a noncentral A must contain a noncentral Lie ideal of R. The main theorem is the additive version of [\textit{I. N. Herstein}, J. Algebra 83, 26-32 (1983; Zbl 0514.16001)], and its proof both uses the results in and follows the general approach of this paper of Herstein. The importance of the main theorem here lies in the fact that it is essentially the first result for subgroups and it implies and genuinely extends almost all of the previous results in this area, which before Herstein's paper, considered only invariant subspaces or invariant subalgebras. The complications resulting from the restriction to additive subgroups are nontrivial, and the main problem is overcome in Lemma 4 with a very clever and nice computation.
scientific article; zbMATH DE number 4002294
scientific article