Let $R$ be a $M\times N$ matrix with rational entries, Is $|(R\mathbb{Z}^N)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M)/\mathbb{Z}^N|$?

Question

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.)

Answer 1

Put $R$ in 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.