mathoverflow.net/users/112259/chuaks this sequence of totally real generalised golden ratios also more than answers my question and is very elegant. Is there any literature on this sequence that you can point me to. I would have ticked your answer as the answer had Henri Cohen not answered first, and it seems I cannot tick two answers. Strictly, Cohen's answer is a complete answer to what I asked but the advantage of your sequence is that as the desired length of sequence is increased so the sequence can be extended. This potentially has useful applications.

Yes I believe your answer. Clearly some condition would be required on R. I wonder if products of free modules are locally free for Noetherian rings?

I don't agree that every element of $R$ is annihilated by $e_1-e_{i+!}$ for sufficiently large $i$.

@AndréHenriques You may well be right, but I have not figured out how to extract a filtered colimit from a resolution.

@AndréHenriques No I haven't: I will take a look at this and report back.

@BenjaminSteinberg It seems that Stefan Waner is assuming $G$ is a compact Lie group in the paper you put forward which gives me some reservations about this being the right reference. This does certainly address the case $G$ is finite!

I am interested in the general case although a reference that covers the case when $G$ is finite could starting point.

Thanks Sam this is a very clear answer!!