Exercise 1, § 2

(a) Show that in the Noetherian polynomial ring $A = \mathbf{Z}[X]$ in one indeterminate over $\mathbf{Z}$, the ideal $m = 2A + AX$ is maximal and that the ideal $q = 4A + AX$ is m-primary, but is not equal to a power of m.

(b) In the ring $B = \mathbf{Z}[2X, X^2, X^3] \subset A$ which is Noetherian, show that the ideal $p = 2BX + BX^2$ is prime, but that $p^2$ is not p-primary, although its radical is equal to p.

(c) Let K be a commutative field and A the quotient ring of $K[X, Y, Z]$ by the ideal generated by $Z^2 - XY$; let x, y, z be the canonical images of X, Y, Z in A. Show that $p = Ax + Az$ is prime, that $p^2$ is not primary and that $p^2 = a \cap b^2$ is a primary decomposition of $p^2$, where $a = Ax$ and $b = Ax + Ay + Az$.