Exercise 1, § 20

Let K be a commutative field. Let A be the K-algebra with basis consisting of the unit element and two elements $e,f$ such that $e^2=e,ef=f$, and $f e=f^2= 0$. a) Prove that the regular and coregular representations of A are not isomorphic (calculate Tr$_{A/K}(e)$ and Tr$_{A^o/K}(e)$).

b) Prove that the regular representation contains a subrepresentation that is isomorphic to neither a subrepresentation nor a quotient representation of the coregular representation. Moreover, there exists a simple representation of A that is not isomorphic to a subrepresentation of the regular representation.