On invariant additive subgroups

Abstract

Suppose thatR is a prime ring with the centerZ and the extended centroidC. An additive subgroupA ofR is said to be invariant under special automorphisms if (1+t)A(1+t)−1 ⊆A for allt ∈R such thatt 2=0. Assume thatR possesses nontrivial idempotents. We prove: (1) If chR ≠ 2 or ifRC ≠C 2, then any noncentral additive subgroup ofR invariant under special automorphisms contains a noncentral Lie ideal. (2) If chR=2,RC=C 2 andC ≠ {0, 1}, then the following two conditions are equivalent: (i) any noncentral additive subgroup invariant under special automorphisms contains a noncentral Lie ideal; (ii) there isα ∈Z / {0} such thatα 2 Z ⊆ {β 2:β ∈Z}.