Exercise 12, § 7

Let $A$ be a noetherian ring, $a$, $b$ and $c$ three ideals of $A$ such that $c$ is contained in the radical of $A$. a) Now prove that if one has the inclusion $\overline{b + c.a} \subset \overline{a}$, then one has $\overline{b} \subset \overline{a}$. b) Deduce that if $A$ is local, and if one has $b \subset a$ and $\overline{b} = \overline{a}$, there exists at least one ideal $b_1 \subset b$ such that $\alpha) \overline{b_1} = \overline{a}$, $\beta)$ $b_1$ is minimal for this property, that is to say that if $b'_1 \subset b_1$ and $\overline{b'_1} = \overline{b_1}$, then $b'_1 = b_1$.