$D\neq (1)$ is an ideal in ring $A\neq0$. Now $A$ has a maximal ideal $I$ such that $D\subseteq I. \ $
Now consider the following set $$S=\{I_i \ |\ I_i\leq A, D\subseteq I_i\}$$ $\mathscr{C}$ is a chain of $S. \ $ Prove that for $C_i\in \mathscr{C}$ and $C=\bigcup C_i $
$$ C\in \mathscr{C}$$ This is mentioned in my book. Can someone help me how can I start this proof. Thanks