I've found several papers studying random matrices over finite fields $\mathbb{F}_q$, but none dealing with matrices over the rings $\mathbb{Z}_m\equiv\mathbb{Z}/m\mathbb{Z}$ for general $m\in\mathbb{Z}_+$. Does someone know any such reference?
In particular, the following question is what I had in mind: Suppose that a $K\times N$, $K\leq N$, matrix is drawn at random, with independent uniformly distributed entries from $\mathbb{Z}_m$. What is the probability (or a non-trivial upper bound on it) that its $K$ rows are linearly dependent?
I suppose such questions have been studied before, but I couldn't find relevant references...