The valuation map of the field, ${\operatorname{val}}\colon K^{\times}\to\mathbb{Z}$, induces a valuation map on $\mathbb{T}(K)$, also denoted ${\operatorname{val}}\colon\mathbb{T}(K)\to N$, by the identifications $\mathbb{T}(K)=\operatorname{Hom}(M,K^{\times})$ and $N=\operatorname{Hom}(M,\mathbb{Z})$.