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Let $0\leq e<e^\prime$ be integers. Now suppose $N$ is the number of $n\times n$ matrices over the ring $R:=\mathbb{Z}/p^{e^\prime}\mathbb{Z}$ (where $p$ is prime) such that $(p^eR)^n\subseteq\mathrm{im}(A)$. I want to find a lower bound say $\ell (e,e^\prime)$ on $N$ such that $$\lim_{e\to\infty} \frac{\ell (e,e^\prime )}{p^{n^2e^\prime}}=1.$$ Essentially I am trying to show that the probability that a matrix satisfies the above property tends to $1$ as $e\to\infty$. I think this result has been shown in some paper by Friedman-Washington, but I don't have access to that paper. Does anyone has any idea how to go about for getting such a lower bound?

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Eduardo Friedman and Lawrence C. Washington, On the distribution of divisor class groups of curves over a finite field, Théorie des nombres (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 227–239. –  Dietrich Burde Feb 10 at 19:40

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