Exercise 1, § 1

Show that if $A$ is an integral domain then the polynomial ring $A[X_i]_{i \in I}$ is not a principal ideal domain when one of the following conditions holds : 1) $\operatorname{Card}(I) \geq 2$; 2) $A$ is not a field (consider the ideals $(X_i) + (X_j)$ for $a \neq \beta$. and $(a) + (X_i)$ where $a \neq ()$ is a noninvertible element of $A$).