Exercise 13, § 8

Suppose that A is left and right noetherian. Now prove that the following conditions are equivalent: α) $\mathrm{dh}(A) \leq 2$. β) For all integers $p, q$ and every a-linear mapping $u : A^p \to A^q$, the kernel of $u$ is a projective a-module. γ) The dual of every finitely generated a-module is projective.