Element of a Group Ring
Let . Then, the question is does hold, for an arbitrary ring?
Clearly, it is enough to show that , that is, .
So, according to my notes, , which I understand perfectly well. However, apparently this is just because . I don't get why that is true!
My thought is perhaps that the action of on elements of a (finite) normal subgroup perhaps induces a unique element of ? That is to say, for , then for ?
Thanks in advance!
Originally Posted by NonCommAlg
That's much easier than I was anticipating. I do have one further question - why is RG a Free Ring? According to Wiki, Group Rings are Free Modules, things I know next to nothing about, but Free Rings are not mentioned...
by "free ring" here i meant a free R-module which also has a structure of a ring. (it would be an algebra if R was commutative) so it's not anything official. i just made it up!
Originally Posted by Swlabr