Exercise 4, § 20

Keep the assumptions of Exercise 3.

a) Prove that if A is a Frobenius ring (VIII, p. 53, Exercise 9), then we have Card(I$_{\alpha}$) $=$ Card(I$_{\sigma(\alpha)}$) for every $\alpha \in R$. Also prove the converse.

b) An algebra of finite degree over a commutative field is Frobenius if and only if its regular representation is isomorphic to its coregular representation (use Exercise 3, c)).