Homology of the Torus over a Module

Question

The homology groups $H_n(T,\mathbb{Z})$ for the torus $T$ are

$\mathbb{Z} \oplus \mathbb{Z}$ for $n=1$,

$\mathbb{Z}$ for $n=0,2$.

Zero otherwise.

Question: Changing the coefficients in $\mathbb{Z}$ for coefficients in a $R$ free module, how do the homology groups look like? Wikipedia says "All of the constructions go through with little or no change". However, I can not follow why is this statement so clear. Could someone explain why is this? And how the homology groups could be computed?

Thanks.