augmentation ideal and $\rho$-isotopic spaces

These are two questions regarding Rubin`s paper – Global units and Ideal class groups in Inventiones 1987. The two questions are from section 3.

Let $ p$ be a rational prime and $ K \subset E \subset F$ , where $ F$ is an abelian extension of $ K$ containing the Hilbert Class Field of $ K$ , with $ G=Gal(F/K)$ . Let $ \rho$ be an irreducible $ \mathbb{Z}_p$ -representation of $ \Delta = Gal(F/E)$ . For any $ \mathbb{Z}[\Delta]$ -module $ M$ , write $ M^{\rho}$ for $ \rho$ -eigenspace of $ \hat{M}= \lim M/p^n M$ . Explicitly, $ M^{\rho}= e_{\rho}\hat{M}$ , where $ e_{\rho}$ is the $ \rho$ -idempotent. Let $ W$ be a submodule of $ (O_F^*)^{\rho}$ such that $ (O_F^*)^{\rho}$ has no $ \mathbb{Z}_p$ torsion. Let $ N=p^n$ .

Since $ \rho \ne 1$ , I have two questions:

1) Why is $ (\mathbb{Z}/N\mathbb{Z})[G]^{\rho}$ contained in the augmentation ideal of $ (\mathbb{Z}/N\mathbb{Z})[G]$ ? I couldn`t understand the action of $ \rho$ on a $ (\mathbb{Z}/N\mathbb{Z})[G]$ -module $ M$ .

2) Let $ V=W/W^N$ . Why is $ O_K^* \cap V = 0$ ?