What is the free abelian group on $M \times N$ where $M,N$ are modules. The free abelian group (equivalently the free $\mathbb{Z}$-module) $F(M \times N)$ is 
defined as the set of all linear combinations of elements of $M \times N$,

$$F(M \times N) = \{n_1x_1 + \cdots +n_kx_k : n_i \in \mathbb{Z}, x_i \in M \times N, k \in \mathbb{Z}_{\ge0}\}$$  

Wouldn't these linear combinations already be in $M \times N$?  How is the free abelian group different from $M \times N$?
No. For instance, suppose you took the $\mathbb{Z}$-module $\mathbb{Z}/2\mathbb{Z}=M$. Then, in $M^2$ you have that $2(1,0)=(0,0)$ but in $F(M^2)$ one has that $2(1,0)\ne (0,0)$. The relations that hold amongst elements of $M^2$ DO NOT hold for elements of $F[M^2]$.
A right $\mathbb Z$-module is also a left $\mathbb Z$-module, if that helps.