In Iwasawa theory, one of the fundamental results is the following structure theorem for finitely generated modules over the ring $\Lambda = \mathbf{Z}\_p[[T]]$.



> 
> If $M$ is a finitely generated torsion $\Lambda$-module, then there is a pseudo-isomorphism
> $$M \to \left( \bigoplus\_{i=1}^s \Lambda/(p^{n\_i}) \right) \oplus \left( \bigoplus\_{j=1}^t \Lambda/(f\_i) \right) $$
> where the $f\_i \in \Lambda$ are distinguished polynomials.
> 
> 
> 


**My question is the following:** are the numbers $s$ and $t$ in the above theorem uniquely determined by $M$? That is, is the number of direct summands in the above decomposition uniquely determined by $M$?


As to *why* I'm asking this, I'm in a situation where I have a finitely generated torsion $\Lambda$ module $M$ and I happen to know that $M$ is generated by $n$ elements $x\_1, \dots, x\_n$. (**EDIT**: I also know that the elements $\{x\_1,...,x\_n\}$ form a *minimal* generating set, so no strict subset of $\{x\_1,...,x\_n\}$ generates all of $M$.) I'd then like to use this to conclude that there are $n$ summands in the direct-sum composition above (i.e: that $s+t=n$). To do this, however, one needs to know that $s$ and $t$ are uniquely determined by $M$, which is why I'm asking this question.


Any tips would be appreciated. Thanks!



Structure theorem for finitely generated $\Lambda$-modules - uniqueness part
By Nakayam's lemma, since $M$ is finitely generated, $x\_1,\dots, x\_n$ generate $M$ if and only if they generate $M/ (p, T)M$ (where $(p,T)$ is the maximal ideal of the local ring $\mathbb Z\_p[[T]]$.)


So $x\_1,\dots, x\_n$ are a minimal generating set if and only if they are a basis of $M/ (p,T)M$, and thus in this case $n$ is the rank of $M/(p,T)M$.


If your pseudo-isomorphism were an isomorphism, you'd be done at this point, as that rank would be $s+t$. However, for a general torsion Iwasawa module, we can have, for example $s=t=0$ with $M$ and thus its minimal generating set nonempty, as happens when $M$ is finite.


So you need some additional condition on $M$.


