# How to turn a group ring $R(G)$ into a ring?

Let $R(G)$ be a given abelian group ring. Any abelian group ring is isomorphic to an abelian ring. I know how to express (isomorphism) some group rings as a ring. But I wonder if there is a general method for finding how to write a given abelian group ring as an abelian ring ? I mean a method that is not trial and error. Also it must be efficient and always halt to the correct answer.

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Are you saying something like $R[C_n]\cong R[x]/(x^n-1)$ and trying to see if there is a general way to describe the group ring of an abelian group in general? –  Alex Youcis Feb 10 '13 at 20:31
Yes but even more than that. As an example $R[C_4]\cong R[x]/(x^4-1)$ is solved by letting $x=(-1,i)$. So I want a way to know what $x$ is too. Not just the $\cong R[x]/(x^n-1)$ part. But also the $x=...$ part. –  mick Feb 10 '13 at 20:46
What's an «abelian group ring»? The group ring of an abelian group? There is some usefulness in saying abelian for groups and commutative for rings... Also, it would not hurt if you made explicit what you mean by group ring, as there are at least two different meanings for that term. –  Mariano Suárez-Alvarez Feb 10 '13 at 21:14
I have no idea at all what you are asking. What do you mean by "turning a group ring into a ring"? It's a ring already. –  Derek Holt Feb 10 '13 at 21:35
I too am curious about what is actually meant by this question. –  Tobias Kildetoft Feb 11 '13 at 15:08

For the case of rational group algebras, one can use the package Wedderga for the GAP system, for example:

gap> LoadPackage("wedderga");
...
gap> QG:=GroupRing(Rationals,CyclicGroup(4));
<algebra-with-one over Rationals, with 2 generators>
gap> WedderburnDecomposition(QG);
[ Rationals, GaussianRationals, Rationals ]
...
gap> QG:=GroupRing(Rationals,ElementaryAbelianGroup(4));
<algebra-with-one over Rationals, with 2 generators>
gap> WedderburnDecomposition(QG);
[ Rationals, Rationals, Rationals, Rationals ]
...
gap> QG:=GroupRing(Rationals,DirectProduct(CyclicGroup(4),CyclicGroup(32)));
<algebra-with-one over Rationals, with 7 generators>
gap> WedderburnDecomposition(QG);
[ Rationals, GaussianRationals, Rationals, CF(32), CF(16), CF(8),
GaussianRationals, Rationals, CF(32), CF(16), CF(8), GaussianRationals,
GaussianRationals, CF(32), CF(16), CF(8), GaussianRationals,
Rationals, CF(32), CF(16), CF(8), GaussianRationals ]


The function WedderburnDecomposition returns a list of simple algebras whose direct sum is isomorphic to the group algebra given as input.

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