Exercise 5, § 2

Let $K$ be a field; in the polynomial domain $K[X, Y]$ in two indeterminates consider the subrings $A = K[X^4, Y^4], A' = K[X^4, X^3Y, XY^3, Y^4]$; $A$ is Noetherian and $A'$ is a finite $A$-algebra. Let $p = AY^4$, which is a prime ideal of $A$; show that $m = A'X^4 + A'X^3Y + A'XY^3 + A'Y^4$ is an (immersed) prime ideal associated with the ideal $pA'$ of $A'$ but does not lie above $p$ (note that $m$ is the annihilator of the class of $X^2Y^6$ in the ring $A'/pA'$) (cf. § 1, Exercise 26 (d)).