§ Ring of power series with infinite positive and negative terms

If we allow a ring with elements xix^i for all <x<-\infty < x < \infty, for notation's sake, let's call it R[[[x]]]R[[[x]]]. Unfortunately, this is a badly behaved ring. Define Si=xiS \equiv \sum_{i = -\infty}^\infty x^i. See that xS=SxS = S, since multiplying by xx shifts powers by 1. Since we are summing over all of Z\mathbb Z, +1+1 is an isomorphism. Rearranging gives (x1)S=0(x - 1)S = 0. If we want our ring to be an integral domain, we are forced to accept that S=0S = 0. In the Barvinok theory of polyhedral point counting, we accept that S=0S = 0 and exploit this in our theory.