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.

Question:

(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 $|(R\mathbb{Z}^N)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M)/\mathbb{Z}^N|$?