It is well-known that there exist Groebner bases for ideals in polynomial ring $\mathbb Q[x]$ which can be found algorithmically Moreover, I don't think it is hard to show that there exist Groebner bases for ideals in $\mathbb Z[x]$. But I am having trouble defining Groebner Bases for submodule of free $\mathbb Z[x]$-modules and showing Groebner bases exist.

So my question is how do we define Groebner bases for $\mathbb Z[x]$-modules and are we able to find it algorithmically?