Exercise 1, § 17

Let B be an algebra of finite degree over the commutative field K, and let $b$ be an element of B.

a) If $b$ is nilpotent, then we have Tr$_{M/K}(b) = 0$ for every B-module M of finite dimension over K. Deduce that if $b$ belongs to the radical of B, then we have Tr$_{M/K}(bc) = 0$ for every $c\in B$.

b) Prove that if Tr$_{S/K}(bc) = 0$ for every simple B-module S and every $c\in B$, then $b$ belongs to the radical of B (use the density theorem).