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

Let $(R\mathbb{Z}^N+\mathbb{Z}^M)=\{R\Lambda+\Lambda'|\Lambda\in\mathbb{Z}^N,~\Lambda'\in\mathbb{Z}^M\}$ which forms a group under addition.

Note that $(R\mathbb{Z}^N+\mathbb{Z}^M)$ has a subgroup $\mathbb{Z}^M$, and hence we can construct the quotient group $(R\mathbb{Z}^N+\mathbb{Z}^M)/\mathbb{Z}^M$. Similarly, we have the notion of $(R^T\mathbb{Z}^M+\mathbb{Z}^N)/\mathbb{Z}^N$

Question:

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

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