The group-algebra of an abelian group is commutative, so we can consider the spectrum of this algebra. Are there any information about the abelian group that we can obtain from such considerations? That is to say, could we study abelian groups by considering the spectrum and the scheme of its group-algebra?  
Since I know nothing about the subject, any reference is mostly welcomed. Thanks in advance.  
P.S. I also posted in mathmatics stack exchange
[Here.](https://math.stackexchange.com/questions/308939/commutative-ring-and-its-group-algebra-and-abelian-group-algebra-as-a-commutati)