I'm using the following result in a computer science paper:


Let $V$ be a submodule of $(\mathbb{Z}/q\mathbb{Z})^n$ (n-tuples with addition and multiplication mod $q$). Let


$$V^\perp = \{u \in (\mathbb{Z}/q\mathbb{Z})^n : \forall v \in V \quad v \cdot u = 0\}$$


where $v \cdot u = v\_1 u\_1 + \ldots + v\_n u\_n \text{ mod } q$.


Propositon: $(V^\perp)^\perp = V$.


I can't find any reference with this exact result even though I'm quite sure it is basic, mostly because I'm not from the mathematics community. I'd be glad to have a reference for $q$ prime, even though I'd rather have the more general result. Do any of you know a good reference for this result? Thank you very much.



Looking for a reference: double orthogonal complement in $(\mathbb{Z}/q\mathbb{Z})^n$
As explained in [this answer](https://mathoverflow.net/a/81938/42091), the property $V=(V^\perp)^\perp$ holds for submodules $V$ of $(\mathbb Z/q\mathbb Z)^n$ because you can use Smith normal form to find a suitable generating matrix for $V^\perp$. If you just want a reference for the result, I can only suggest the following paper:


D. Wilding, M. Johnson, M. Kambites. Exact rings and semirings.
*Journal of Algebra*, 388 (2013), 324–337; [arXiv:1212.5358](http://arxiv.org/abs/1212.5358).


When writing this paper we faced the same problem as you. That $(V^\perp)^\perp=V$ follows quite easily from known results, but we could not find it written down explicitly in this formulation.


