Exercise 6, § 7

Let A be a quasi-simple ring.

a) An A-module is generating if and only if its dual is nonzero.

b) Let $\mathfrak{a}$ be a left ideal of A, and let D be the ring End$_A(\mathfrak{a})$. Prove that $\mathfrak{a}$ is a finitely generated projective right D-module and that the mapping $a\mapsto a_{\mathfrak{a}}$ from A to End$_D(\mathfrak{a})$ is bijective. The ring D is quasi-simple if and only if the A-module $\mathfrak{a}$ is projective and finitely generated.

c) Prove that a quasi-simple ring that contains a minimal left ideal is simple.