### Abstract

We study the problem of constructing complements to the valuation ring of a valued field, which are $k$-algebras over the residue field $k$.

These $k$-algebras have many important arithmetic properties, and are closely related to truncation integer parts.

We work out the case of henselian fields and study the canonical integer part $\Neg(F)\oplus\Z$ of any truncation closed subfield $F$ of the field of power series $k((G))$, where $\Neg(F):=F\cap k((G^{<0}))$. In particular, we prove that $k((G^{<0}))\oplus\Z$ has (cofinally many) prime elements for any ordered divisible abelian group $G$.

Addressing a question in a paper of Berarducci, we show that every truncation integer part of a non-archimedean exponential field has a cofinal set of irreducible elements. Finally, we apply our results to two important classes of exponential fields:exponential algebraic power series and exponential-logarithmic power series.