Let $R$ be a $M\times N$ matrix with rational entries, $R\mathbb{Z}^N$ be the image of $\mathbb{Z}^N$ under R.


Consider a equivalent relation on $R\mathbb{Z}^N$ defined by
$a\sim b$ if $a-b\in \mathbb{Z}^M$ for any $a,~b\in R\mathbb{Z}^N$.
Denote the set of equivalent classes as $(R\mathbb{Z}^N)/\mathbb{Z}^M$.
Similarly, we have the notion of $(R^T\mathbb{Z}^M)/\mathbb{Z}^N$.
Both $(R\mathbb{Z}^N)/\mathbb{Z}^M$ and $(R^T\mathbb{Z}^M)/\mathbb{Z}^N$ form groups under addition.


Questions:


(1) Is $(R\mathbb{Z}^N)/\mathbb{Z}^M$ isomorphic to $(R^T\mathbb{Z}^M)/\mathbb{Z}^N$?


(2) If (1) is not true, Is the cardinality $|(R\mathbb{Z}^N)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M)/\mathbb{Z}^N|$?


(This is posted on both Math Overflow and Math Stack Exchange.)



Let $R$ be a $M\times N$ matrix with rational entries, Is $|(R\mathbb{Z}^N)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M)/\mathbb{Z}^N|$?
Put $R$ in [Smith normal form](http://en.wikipedia.org/wiki/Smith_normal_form). While this is usually defined for integer matrices, for a rational matrix $R$, we may write $R = P D Q$, where $P$ and $Q$ are in $GL\_M(\mathbb{Z})$ and $GL\_N(\mathbb{Z})$ respectively, and $D$ is a diagonal matrix with diagonal entries $d\_1, d\_2,\dotsc\in\mathbb{Q}$ such that $d\_{i+1}$ is an integer multiple of $d\_i$ for $i = 1, 2, \dotsc$.


Now note that replacing $R$ by $D = P^{-1}RQ^{-1}$ does not change the isomorphism class of $R\mathbb Z^N/\mathbb Z^M$ or $R^T\mathbb Z^M/\mathbb Z^N$ (since $Q^{-1}\mathbb Z^N = \mathbb Z^N$ and $P^{-1T}\mathbb Z^M = \mathbb Z^M$).


Since $D$ is diagonal, $D\mathbb Z^N/\mathbb Z^N$ and $D^T \mathbb Z^M/\mathbb Z^M$ are both $\bigoplus\_i (d\_i \mathbb Z/\mathbb Z)$, and therefore isomorphic.


