Ideals are linearly ordered in evaluation rings

Question

How do I prove the fact that for any valuation ring $V$ the ideals are totally ordered under inclusion?

Answer 1

Hint:

• Prove first the principal ideal in $V$ are totally ordered by inclusion: for this, let $a,b\in V$. Show that if $Va\not\subset Vb$, then $Vb\subset Va$.
• Deduce that, if $\mathfrak a$ and $\mathfrak b$ are two ideals in $V$, if $\mathfrak a\not\subset \mathfrak b$, then $\mathfrak b\subset \mathfrak a$ (take $a\in\mathfrak a$, $\;a\notin\mathfrak b$. Show that, for any $b\in\mathfrak b$, $b\in Va$).