Exercise 20, § 9

Let A be an algebra over a commutative field K such that $\mathfrak{R}(A)$ is a nil ideal and that the algebra $A/\mathfrak{R}(A)$ is isomorphic to a finite product of matrix algebras over K. Prove that there exists a subalgebra S of A such that the K-vector space A is the direct sum of S and $\mathfrak{R}(A)$ (use Exercises 18 and 19 above, as well as Exercise 19 of VIII, p. 94).