Let $p$ be a prime and $M$ is a finitely generated $ \mathbb{Z}\_{p}[[T]] $ module. Suppose $M[p]$ denotes the $p$-torsion of $M$. Then $M[p]$ and $M/(p)$ are both $ F\_{p}$ vector spaces. So we can talk of their dimensions. Now what can we say about the rank of $M$ from looking at the dimensions of $M[p]$ and $M/(p) ?$



Rank of a $ \mathbb{Z}_{p}[[T]] $ module
Not much : take $M=\mathbb{Z}\_p[[T]]^r$. Then $M[p]$ is zero, and $M/pM=\mathbb{F}\_p[[T]]^r$ is infinite-dimensional...


