I am currently trying to read the paper Constructions of difference sets by Applebaum et al. Unfortunately, there are some very elaborate calculations. For example, in example 1.13 in the group ring $\mathbb{Z}K$ $$ (1+X+Y-XY)(1+X^2+Y^2+X^2Y^2)(1+X^3+Y^3-X^3Y^3), $$ where $K=C_4^2=\langle X,Y \rangle$ is a product of cyclic groups. Is there a computer algebra system that can help me with this? I am not that familiar with it.
The result should be $4(1+X^2+Y^2+X^2Y^2)$.
Indeed, multiplying out and taking the relations $X^4=Y^4=1$ we obtain $$ 4(X^2Y^2 + X^2 + Y^2 + 1) $$ This can be computed with any CAS, and even by hand. Examples of CAS are Maple, Mathematica, Singular, Gap, Reduce, Magma and many others. For a list see also wikipedia.