Valuation ring of finite Krull dimension whose every non-maximal ideal is principal

Question

Let $(R, \mathfrak m)$ be a Valuation ring of finite Krull dimension such that every non-maximal ideal i e. every ideal which is not $\mathfrak m$, is principal. Then is $R$ Noetherian i.e. a discrete valuation ring ?

Answer 1

If $\mathfrak{m}=0$, then $R$ is a field. Otherwise, let $r\in\mathfrak{m}$ be nonzero and let $I=r\mathfrak{m}$. Then $r\not\in I$ so $I$ is a non-maximal ideal, and hence is principal. But multiplication by $r$ is an isomorphism of $R$-modules from $\mathfrak{m}$ to $I$, so this means $\mathfrak{m}$ is principal as well. Thus every ideal in $R$ is principal and $R$ is Noetherian.