Finitely Generated Modules over Quotient of a DVR. Let $(R,t)$ be a DVR with uniformizing parameter $t$ and let $M$ be a finitely generated module over $R/(t^n)$. Then $M\cong \bigoplus_r R/(t^r)$.
Question: How many summands of type $R/(t^r)$, where $1\leq r\leq n$ appear in the above decomposition? Can we say anything more than the fact that there are finitely many? Any formula to compute the above, however unsatisfactory is appreciated.
Thanks

Finitely many.  That's all that can be said without further information on $M$.  
(To say a little more: a finitely generated module over $R/(t^n)$ is just a finitely generated module over $R$ which is killed by $t^n$.  Thus the theory of invariant factors / elementary divisors over the PID $R$ applies.  What it says is that $M$ is isomorphic to a finite direct sum of cyclic $t^n$-torsion modules.  Each such guy is of the form $R/(t^r)$ for some $1 \leq r \leq n$.  For each given $r$ you can have any finite number of summands of this form.)
