Answer to Question #18666 in Abstract Algebra for john.george.milnor

Question

Let J be a nil ideal in an algebra R over a field k of characteristic 0, and let G be the group 1 + J ⊆ U(R). For any y ∈ G, let y^k := {y^α : α ∈ k}. If y <> 1, show that y^k is a subgroup of G isomorphic to the additive group k.