If $M=\mathbb{Z}\oplus \cdots \oplus \mathbb{Z}$ is a finitely generated free $\mathbb{Z}$-module of rank $n$, and if we have $n$ vectors $v_1,\ldots,v_n\in M$ which are $\mathbb{Z}$-independent, then the subgroup $\mathbb{Z}v_1 + \cdots + \mathbb{Z}v_n$ is of finite index which can be computed using determinant (with absolute value) of $n\times n$ integer matrix, whose $n$ rows are $v_1,\ldots,v_n$.
Now consider the situation where module is over ring bigger than $\mathbb{Z}$:
More specifically, let $M=\mathbb{Z}[\omega]\oplus \mathbb{Z}[\omega]\oplus \mathbb{Z}[\omega]$ and consider following vectors in it:
[Edit: $\omega$ is primitive cube root of $1$ in $\mathbb{C}$.]
$$v_1=(1,1,1), \,\, v_2=(1,\omega,\omega^2), \,\, v_3=(1,\omega^2,\omega).$$ Let $N=\mathbb{Z}[\omega]v_1 + \mathbb{Z}[\omega]v_2 + \mathbb{Z}[\omega]v_3$.
Q.1 How can we decide if index of additive subgroup $N$ in $M$ s finite or not? and how can we compute if it is finite?
Q.2 What is the relation of $[M:N]$ with determinant of matrix with rows $v_1,v_2,v_3$?
