Let $R$ be a principal ideal domain and consider an infinite strictly decreasing chain of ideals in $R$, say $I_1\supset I_2\supset \dots$. Show that $\cap_{i=1}^{\infty} I_i =(0)$
(I took down my first attempt, and heres where I am so far)
My attempt at a proof:
Since $R$ is a prinicpal ideal domain, every ideal in $R$ is a principal ideal. This implies that for an infinite strictly decreasing chain of ideals in $R$, $I_1\supset I_2\supset \dots$. each $I_i$, where $i\in\mathbb{N}$, is generated by a single element of $R$, $a_i$.