Exercise 12, § 2

In a ring A let $r(a)$ denote the radical of an ideal a. (a) Show that, for three ideals $a, b, c$ of A,

(b) Give an example of an infinite sequence of ideals $(a_n)$ of A such that $r(\bigcap_n a_n) \neq \bigcap_n r(a_n)$. (c) Give an example of a principal ideal $a$ such that $r(a)$ is not finitely generated and there exists no integer $n > 0$ such that $(r(a))^n \subset a$. (Consider the ring of a non-discrete valuation of height 1.)