Answer to Question #18082 in Abstract Algebra for Melvin Henriksen

Question

For any ring R with Jacobson radical J, we have an exact sequence of groups
(∗) 1→ 1 + J → U(R) → U(R) → 1 (where R = R/J),
induced by the projection map π : R → R. Show that this sequence splits if R is a commutative rad-nil Q-algebra, or R is a commutative artinian ring.