Exercise 27, § 2

(a) Let $A$ be a ring and $M$ a finitely generated faithful $A$-module. Show that, if $a, b$ are two ideals of $A$ such that $aM = bM$, the radicals of $a$ and $b$ are equal (use no. 2, Corollary 2 to Proposition 4).

(b) Deduce from (a) that, if in an integral domain $A$ $p$ is a finitely generated prime ideal, then $p^m \neq p^n$ for $m \neq n$.

(c) Let $K$ be a field and $A$ the polynomial ring $K[X, Y]$ in two indeterminates. Let $a$ be the ideal $AX^4 + AX^3Y + AXY^3 + AY^4$, $b$ the ideal $a + AX^2Y^2$ of $A$; show that $b \neq a$ and $b^2 = a^2$.