Module isomorphism and trace

Module isomorphism and trace

I was reading the document "The Different Ideal"
http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/different.pdf and
there is a part on page 9, which I don't quite get.
We have a number field $K$ and its ring of algebraic integers $O_K$. Say
$\mathcal{P}$ is a prime ideal of $O_K.$ Take an element $\pi \in
\mathcal{P} \backslash\mathcal{P}^2 $. Then, $(\pi^i)$ is divisible by
$\mathcal{P}^i$, but not by $\mathcal{P}^{i+1}$, so $\mathcal{P}^{i} =
(\pi^i) + \mathcal{P}^{i+1}$. Therefore, $O_K/ \mathcal{P}$ is isomorphic
to $\mathcal{P}^i / \mathcal{P}^{i+1}$ as $O_K$-module isomorphism by $x
\mod \mathcal{P} \rightarrow \pi^i x \mod \mathcal{P}^{i+1}$.
I understand until here and the following part is where I am stuck on..
This $O_K$ module isomorphism comments with multiplication by $y$ on both
sides, so
$$ Tr (m_y: \mathcal{P}^i / \mathcal{P}^{i+1} \rightarrow \mathcal{P}^i /
\mathcal{P}^{i+1} ) = Tr (m_y: O_K / \mathcal{P} \rightarrow O_K /
\mathcal{P}). $$
I would greatly appreciate an explanation! Thank you very much!