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Crossposted from math.stackexchange, since I'm not getting any answer and I think the question is suitable here.

Given a generating set of a $\mathbb{Z_k}$-module $M \subseteq {\mathbb{Z}_k}^n$, is there an (efficient) known algorithm to compute a generating set of {$u \in {\mathbb{Z}_k}^n : \forall v \in M \quad v \cdot u = 0$}? It is quite simple when $k = p^i$ for a prime $p$ (since it becomes a vector space), but I have no clue what to do when $k$ is composite.

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Why is $\mathbb{Z}_k$ a field for $k=p^i$ when $i>1$ ? – tomasz Sep 6 '11 at 19:11

The magic words are "Smith Normal Form". The magic reference is "Integral Matrices" by M. Newman, which discusses many questions of this sort in a lucid way.

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