Exercise 2, § 2

Let $K$ be a commutative field of characteristic $\neq 2$ and $A$ the ring $K[[X, Y]]$ of formal power series in two indeterminates. (a) Show that in $A$ the principal ideal $p = (X^2 - Y^3)$ is prime. (If a product $f(X, Y)g(X, Y)$ of two formal power series is divisible by $X^2 - Y^3$, note first that $f(T^3, T^2) = 0$ or $g(T^3, T^2) = 0$ in the ring of formal power series $K[[T]]$; assuming for example $f(T^3, T^2) = 0$, show first that $f(X, Y^2)$ is divisible by $X - Y^3$ and by $X + Y^3$ and, by considering $f(-Y^3 + X, Y^2)$, prove finally that $f(X, Y^2)$ is divisible by $X^2 - Y^6$.) (b) Let $m$ be the maximal ideal $AX + AY$ of $A$. Show that $p$ is closed with respect to the $m$-adic topology on $A$ (with which $A$ is Hausdorff and complete) but that in the ring $\operatorname{gr}(A)$, $\operatorname{gr}(p)$ is not a prime ideal ($p$ being given the filtration induced by that on $A$).