Answer to Question #22755 in Abstract Algebra for sanches

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.