Exercise 3, § 10

Let A be an Artinian ring such that the ring $A/\mathfrak{R}(A)$ is simple. Prove that A is isomorphic to a matrix ring $\mathbf{M}_r(B)$ over an Artinian local ring B (use Exercises 19 and 20 of VIII, p. 169, as well as Exercise 19 of VIII, p. 94).