Exercise 13, § 2

(a) Let $K$ be a field of characteristic 0 and $n$ the ideal of the polynomial ring $K[X, Y, Z]$ generated by $Y^2 - X^2 - X^3$. Show that $n$ is prime and that the integral domain $A = K[X, Y, Z]/n$ is not integrally closed: if $x, y, z$ are the images of $X, Y, Z$ in $A$, the element $t = y/x$ of the field of fractions $L$ of $A$ is integral over $A$ but does not belong to $A$. Let $A' \supseteq A$ be the subring $K[x, t, z]$ of $L$, which is integral over $A$. Consider in $A$ the prime ideal $p$ generated by $xz - y$ and $z^2 - 1 - x$ and the maximal ideal $q \supset p$ generated by $x, y, z - 1$. Consider on the other hand in $A'$ the maximal ideal $q'$ generated by $x, t + 1$ and $z - 1$; show that $q'$ lies above $q$ but that there exists no prime ideal $p' \subset q'$ of $A$ lying above $p$. (Consider a homomorphism from $A'/pA'$ to an extension field of $K$ and show that the image of $q'$ under such a homomorphism is necessarily 0.)

(b) Let $A'$ be the quotient ring of the polynomial ring $\mathbf{Z}[X]$ by the ideal $n$ generated by $2X$ and $X^2 - X$; let $x$ be the image of $X$ in $A'$; $A'$ is integral over $\mathbf{Z}$ but the number 2 is a divisor of 0 in $A'$. Let $q'$ be the prime ideal of $A'$ generated by 2 and $x - 1$; then $q' \cap \mathbf{Z} = 2\mathbf{Z}$; if $p$ is the prime ideal (0) in $\mathbf{Z}$, show that there exists no prime ideal $p' \subset q'$ contained in $A'$ which lies above $p$.

(c) Let $K$ be a field, $n$ the ideal of the polynomial ring $K[X, Y, Z]$ generated by $X^2, XY, XZ, YZ - Y$ and $Z^2 - Z$, $A'$ the quotient ring $K[X, Y, Z]/n$, $x, y, z$ the images of $X, Y, Z$ in $A'$ and $A$ the subring $K[x, y]$ of $A'$; $A$ is integrally closed in its total ring of fractions and $A'$ is integral over $A$. Consider in $A$ the prime ideal $p$ generated by $x$ and the maximal ideal $q$ generated by $x$ and $y$. Consider on the other hand in $A$ the maximal ideal $q'$ generated by $x, y, z$; show that $q'$ lies above $q$ but that there exists no prime ideal $p' \subset q'$ lying above $p$ (same method as in (a)).

Interpret the examples in (a) and (c) geometrically. \*