Here is one thing I suspect is obvious, but I am not seeing: (Everything is commutative in what I am saying)
Let R[G] be a group ring for some finite group G and suppose that S is a ring which is a R[G]-module as well.
Suppose that there is an element a \in S such that Tr(a) = \sum_{g \in G} ga = 1. Let now x \in S and consider
\sum_{g \in G} (ga)*(g^{-1}x) . I am reading the claim that this is equal to x. Presumably, this is because the sum
is equal to \sum_{g \in G} (ga)*x, but I do not see why... What am I missing?