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? 



Groebner Bases for submodule over polynomial ring with integer coefficients
There is a description of the appropriate Groebner basis algorithm in this book:


Franz Pauer, Andreas Unterkircher.
[Gröbner Bases for Ideals in Laurent Polynomial Rings and their Application to Systems of Difference Equations.](https://doi.org/10.1007/s002000050108)
AAECC 9, 271–291 (1999)


I've implemented it in the single-variable case (in the software Regina) and I've been meaning to implement it in the multi-variable case as well. But I usually get too sad to finish – when I look at how inefficient the algorithm is. Some day I'll have it fully implemented in Regina.


The book is quite well written. I find it easy to read.


