Dear GAP Forum,

I'm very new to GAP, and would appreciate your help to find my way
around it. Given an n-by-m matrix, with entries from the integer
group ring ZG of a free abelian group G, I would like GAP to
calculate the determinants of all k-by-k square sub-matrices (k \leq
min(m,n)), and then define I_k to be the ideal of ZG generated by
these determinants. My problems are:

1) I don't see how to define the ring ZG in GAP - the closest I can
get is a Multivariate Polynomial Ring, or a Univariate Laurent
Polynomial Ring. Can GAP produce group rings?

2) GAP seems to assume that the entries in any matrix are in a field.
Is there any way to avoid this?

3) Once I have my ideal I_k, is there a way of getting GAP to give a
"nice" basis for it, so that I can decide membership? Can GAP
produce Grobner bases?

I'm grateful in advance for any help you can give.

David Cruickshank