Exercise 5, § 1

Let $A = K[X, Y]$ be the polynomial algebra in two indeterminates over a commutative field $K$ and $a$ the maximal ideal $AX + AY$ of $A$. Show that $\operatorname{Supp}(a) = \operatorname{Spec}(A)$ is infinite and $\operatorname{Ass}(a) = \{0\}$. Prove that there is no composition series of a all of whose factor modules are isomorphic to A. (The existence of such a series would imply that a was isomorphic to a module A"; then necessarily $n = 1$, which is absurd.)