Exercise 5, § 20

Let K be a commutative field and A be a K-algebra. We endow $A^*$ with its canonical left A-module structure. For $t\in A^*$, we denote by $\varphi_t$ the mapping $(x, y)\mapsto t(xy)$ from $A\times A$ to K.

a) Prove that this defines a K-linear isomorphism from $A^*$ to the set of K-bilinear mappings $\varphi : A\times A\rightarrow K$ satisfying $\varphi (xy, z) =\varphi (x, yz)$ for $x, y, z\in A$.

b) We say that the element $t\in A^*$ is invertible if the homomorphism $a\mapsto at$ from A to $A^*$ is bijective. Let A be a K-algebra of finite degree. Prove that the following conditions are equivalent:

(i) A is Frobenius (Exercise 4).

(ii) A admits an invertible linear form.

(iii) A admits a linear form whose kernel does not contain any left ideals of A.