Invariant additive subgroups of a certain kind
It is shown here that if R is a prime ring with a non-zero nil right ideal and W is an additive subgroup of R which is invariant under all inner automorphisms of R induced by elements of the form \(1+x\), where x belongs to a nil right ideal of R, then either W is central or W contains a noncommutative Lie ideal J of R. By a theorem of the author and \textit{S. Montgomery} [Pac. J. Math. 42, 117-136 (1972; Zbl 0243.16018)] the Lie ideal J must, in turn, contain a Lie ideal of the form [I,R] where I is a non-zero ideal of R. The result described above extends a result, due to \textit{I. N. Herstein} [Isr. J. Math. 26, 205-208 (1977; Zbl 0343.16006)] in which subrings rather than additive subgroups were considered and which is used in the proof of the present result.
scientific article; zbMATH DE number 4073218