Exercise 6, Appendix 1

We say that a $k$-algebra is primitive if it admits a pseudomodule that is simple and faithful (that is, whose annihilator is reduced to 0).

a) Show that if the algebra $\widetilde{A}$ is primitive, then so is A.

b) If the algebra A is primitive and unital, then the algebra $\widetilde{A}$ is not primitive.

c) Suppose that $k$ is a field and that the algebra A is primitive and not unital. Prove that $\widetilde{A}$ is primitive (prove that if there exists a faithful pseudomodule over A that is not faithful as an $\widetilde{A}$-module, then A admits a right unit element $u$, and deduce a contradiction).