Rank of a $ \mathbb{Z}_{p}[[T]] $ module

Question

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 $M/(p) ?$

Answer 1

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...

Answer 2

It should be noted that the ring $\mathbb{Z}_p[[T]]$ is the Iwasawa algebra of the additive group $\mathbb{Z}_p$, and there is a nice structure theory for finitely generated modules over it. See the linked Wikipedia article, and the references therein, as well as e.g. chapters 7 and 13 of Washington's Introduction to Cyclotomic Fields or chapter 5 of Lang's Cyclotomic Fields (and for a much more general account, Bourbaki's Commutative Algebra ch. VII §4). However, this structure theory works "up to pseudo-isomorphism" i.e. up to so-called "pseudo-null" modules which in this case are exactly the ones with finite cardinality. All subquotients which are finite-dimensional $\mathbb{F}_p$-vector spaces are in this class; on the other hand, the rank is invariant under pseudo-isomorphism. So it is highly unlikely that the dimensions in your question will give you any non-trivial information about the rank of your module, as illustrated by abx's answer.

Edit: When you consider the $\mathbb{F}_p[[T]]$-module structure as you did between two edits, from this structure theory you get

$rank_{\mathbb{Z}_p[[T]]} M = rank_{\mathbb{F}_p[[T]]} (M/(p)) - rank_{\mathbb{F}_p[[T]]} (M[p])$

but not much more, as far as I can see.