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!

minimal. Hopefully that should clarify things. $\endgroup$