$\mathbb{Z}^n/R$ where $R$ is subgroup generated by rows of $n\times n$ integer matrix $M$.

Question

Let $R$ be the subgroup of $\mathbb{Z}^n$ generated by the rows of integer matrix $M$.

I want to show that $det M=0$ implies $\mathbb{Z}^n/R$ is infinite, and also that $det M\neq 0$ implies $|\mathbb{Z}^n/R|=|det M|$.

I know I must use the Smith normal form in some way, as that is what we have been studying.