Exercise 10, § 3

(a) Let $p : A \to B$ be a ring homomorphism. Show that, for every left ideal $a$ of $A$ which is a left annihilator of a subset $M$ of $A$, $\bar{\rho}^{-1}(Ba) = a$

(b) Deduce from (a) an example of a homomorphism $\rho : A \to B$ such that right $A$-module $B$ is not flat but $\bar{\rho}^{-1}(Ba) = a$ holds for every left ideal $a$ of $A$ (cf. § 2, Exercise 17 and Algebra, Chapter VIII, § 3, Exercise 11 and § 2, Exercise 6 and Chapter IX, § 2, Exercise 4).