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 $M/(p) ?$

$\begingroup$ What's $\mathbb Z_p$? Not the $p$adic integers? What do you mean by torsion of a $\mathbb Z_p[[T]]$module? $\endgroup$ – Fernando Muro Dec 1 '13 at 8:34

1$\begingroup$ Wouldn't you want to consider the $\mathbb{F}_p[[t]]$module structure of the reduction, instead of the $\mathbb{F}_p$module structure? $\endgroup$ – S. Carnahan♦ Dec 1 '13 at 9:34

$\begingroup$ @FernandoMuro $ \mathbb{Z}_{p} $ is the padic integers. $\endgroup$ – Suman Dec 1 '13 at 18:12
Not much : take $M=\mathbb{Z}_p[[T]]^r$. Then $M[p]$ is zero, and $M/pM=\mathbb{F}_p[[T]]^r$ is infinitedimensional...
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 pseudoisomorphism" i.e. up to socalled "pseudonull" modules which in this case are exactly the ones with finite cardinality. All subquotients which are finitedimensional $\mathbb{F}_p$vector spaces are in this class; on the other hand, the rank is invariant under pseudoisomorphism. So it is highly unlikely that the dimensions in your question will give you any nontrivial 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.