Answer to Question #18659 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). Show that the map exp : J → G defined by the Taylor series of the exponential function is a one-one correspondence between J and G, with inverse given by the log-function.