Let $K\supset F$ a finite extension of local fields. It means that the valuation $v_K$ extends the valuation $v_F$. We denote with $\pi_K$ and $\pi_F$ the uniformizer parameters and with $\mathcal O_K$, $\mathcal O_F$ the rings of integeters.
Clearly we have the following inclusion of open, compact subgroups:
$$K\supset\ldots\supset\pi_K^r\mathcal O_K\supset\pi_K^{r+1}\mathcal O_K\supset\ldots\supset \{0\}$$
$$F\supset\ldots\supset\pi_F^r\mathcal O_F\supset\pi_F^{r+1}\mathcal O_F\supset\ldots\supset \{0\}$$
Is it true that $\operatorname{Tr}_{K|F}\left(\pi_K^r\mathcal O_K\right)\subset\pi_F^s\mathcal O_F$ for some $s\in\mathbb Z$? Can we say something about such $s$? in other words I wanted to know if the trace map preseves the structure of power of ideals in the local fields.
Thanks in advance