Exercise 5, § 18

Let K be a finite field, $m$ an integer, and A a K-algebra without radical whose elements all have degree $\leqslant m$. Prove that there exists a finite extension L of K such that A is isomorphic to a subalgebra of a product $(\mathbf{M}_r(L))^I$ whose projections are simple algebras (cf. Exercise 2, a)). Also prove the converse.