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A Hahn series over a field $k$ with exponents in an ordered abelian group $\Gamma$ is a formal sum ${\sum }_{\gamma }{a}_{\gamma }{t}^{\gamma }$ whose nonzero exponents form a well-ordered set. Such series form the Hahn series field.

This can be false. In a valued field with value group $\mathbb{Q}$, such as the Hahn series field $k((t^{\mathbb{Q}}))$, the maximal idealof the valuation ring is not principal: if $v(a)=q>0$, an element of valuation $q/2$ belongs to $\mathfrak{m}$ but not to $(a)$. Thus the valuation ring need not be a principal ideal domain.