I defined the product in the group ring by “setting and extending linearly”. A couple of people asked me what I meant by that.
I am referring to the fact that a linear function is determined uniquely by its values on a basis. More precisely, suppose I want to define a linear function between vector spaces, and I have a basis for . Then:
a) Once you choose for each , you will have determined for all , and
b) You may choose for each in any way that you want (as long as they are in ).
In the situation above, the elements (for ) form a basis for . What I said simply means that I am defining for (which is more explicit but messier when written like this).