Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2005 / Algebra / 3 / ii / Solution

Let $J={\bigcup }_n{I}_n$. It contains zero. If $a,b\in J$, both belong to $I_N$ for some common $N$, so $a-b\in I_N\subseteq J$. If $r\in R$, then $ra\in I_N\subseteq J$. These are the ideal axioms, proving $J$ is an ideal.

In a principal ideal domain write $J=(g)$. Since $g\in J$, there is $n_0$ with $g\in I_{n_0}$. For $n\ge n_0$,All inclusions are equalities. Hence the chain stabilizes at $I_{n_0}$. This includes $J=(0)$ and is the ascending chain condition for ideals.