Exercise 4, Appendix 1

Let A be a $k$-algebra and $\mathfrak{a}$ a left ideal of A; we view $\mathfrak{a}$ as a $k$-algebra. Prove that we have $\Re (A)\cap \mathfrak{a}\subset \Re (\mathfrak{a})$ (apply Proposition 7 of VIII, p. 441) and that we have equality when $\mathfrak{a}$ is two-sided (observe that every simple pseudomodule over A is simple over $\mathfrak{a}$ or annihilated by $\mathfrak{a}$). Give an example of a left ideal $\mathfrak{a}$ such that $\Re (A)\cap \mathfrak{a}\not=\Re (\mathfrak{a})$ (take A to be a matrix algebra).