Please give some hints or references for proving the following -

Let $p$ be an odd prime and $\mathbb{Z}_p$ denotes the $p$-adic integers. Suppose $M$ is a finitely generated torsion $\mathbb{Z}_p[[T]]$-module and $\mu(M)=0$. Then $M/pM$ and $M[p]$($p$-torsion points of $M$) are both finite dimensional $\mathbb{F_p}$-vector spaces. How to prove that $\lambda(M)$ is the difference between the $\mathbb{F_p}$-dimensions of $M/pM$ and $M[p]$ $?$

Let $\mathbb{Z}_p$ denotes the $p$-adic integers for a prime $p$. Suppose $M$ is a finitely generated torsion $\mathbb{Z}_p[[T]]$-module such that $\mu(M)=0$. Then $M/pM$ and $M[p]$($p$-torsion points of $M$) are both finite dimensional $\mathbb{F_p}$-vector spaces. How to prove $$\lambda(M) = \dim_{\mathbb{F}_p} M/pM - \dim_{\mathbb{F}_p} M[p] \enspace ? $$