Exercise 2, § 20

Let A be a K-algebra. We view the dual $A^*$ of A as a left A-module.

a) The mapping $E\mapsto E'$ is a surjection from the set of A-submodules of $A^*$ to the set of right ideals of A. When E is such a submodule, the orthogonal of $E'$ in $A^*$ is a submodule of $A^*$ containing E.

b) Prove that the orthogonal in A of the socle of $A^*$ is the radical of A.

c) Prove that the orthogonal in $A^*$ of the socle of $A_d$ is the radical of the A-module $A^*$. Deduce that $A^*$ is a semisimple A-module if and only if the ring A is semisimple.