Show that the union of two ideals need not be an ideal.
Solution
In \(\mathbb Z\), let \[ I=2\mathbb Z,\qquad J=3\mathbb Z. \] Then \(2\in I\subseteq I\cup J\) and \(3\in J\subseteq I\cup J\), but \[ 2+3=5\notin I\cup J. \] Thus \(I\cup J\) is not closed under addition and hence is not an ideal.