Invariant Subgroups in Rings with Involution

Let R be a ring with involution*. In this paper, we study additive subgroups A of R which are invariant under all mappings of the form ϕ x : a → xax*. That is, xAx* ⊆ A for all x ∈ R. Obvious examples of such subgroups A are ideals of R, the set of symmetric elements, and the set of skew-symmetric elements. We will prove that when R is *-prime, these examples are essentially the only ones.