Exercise 5, § 7

Let A be a ring.

a) Let $e$ be an idempotent in A. Prove that if A is primitive (resp. quasi-simple), the same holds for the ring $eAe$ (use Exercise 3, b), Lemma 4 of VIII, p. 113, and Exercise 6, a) of VIII, p. 15).

b) Let $n$ be an integer $\geqslant 1$. Prove that the matrix ring $\mathbf{M}_n(A)$ is primitive (resp. quasi-simple) if and only if A is primitive (resp. quasi-simple) (cf. VIII, p. 114, Example 2).

c) Let B be a ring that is Morita equivalent to A; then B is primitive (resp. quasi-simple) if and only if A is.