Let $R$ be DVR（discrete valuation ring）, M be the only maximal ideal of $R$. We can define discrete valuation on $R$, that is, $$\text{ord}_p(f) = \sup \{ d \in \mathbb N : f \in M^{d} \}$$
Let $t$ be a generator for $M$. On the other hand,arbitrary element of $R$ is generated by unit, that is, for all $ f∈R $, $f = u\cdot t^n$ for some unit $u \in M$ and some non-negative integer $n$.
I want to prove $\text{ord}_p(f)$＝$n$.
I can check this is true when $R＝\Bbb Z$,and p-adic valuation, but cannot prove in general. Thank you for your help.