Lower bound on Indexing set of Formal Laurent Series

Question

For a formal Laurent series defined over a ring R, we require that the indexing set is finitely bounded in the negative direction, or equivalently that the sequence of coefficients of R terminates at some finite negative integer. Which brings me to my question: Why do we need to assume this? I'm guessing that it is related to Noetherian rings, but, to me, that seems more like a convention that we require rather than something that we must assume.

Answer 1

Consider a similar issue: you have a ring $R$ and you consider formal powerseries $ R[\![ t ] \!]$. It is clear we can always multiply them and sum them, now consider a powerseries of the form say $1+x+x^2+\cdots$ and suppose you wanted to "plug in" $x=1+t$. This is readily seen to be problematic, since at each expansion $(1+t)^n$ you would have a trailing $1$, which gives an infinite sum in the constant term that, under usual notions of convergence, doesn't converge. Similarly, consider an infinite negative powerseries $\cdots+t^{-2}+t^{-1}+1$. If you were to multiply this by $1+t+t^2+\cdots$ you would again get infinitely many $1$ terms. Of course you can't "multiply" this say in $\Bbb C$ either; since the series have disjoint regions of convergence (the unit ball and its complement). This problems can be fixed by considering $\mathfrak a$-adic topologies on the ring in question for a suitable ideal $\mathfrak a$.

Answer 2

If you don't assume that you get infinite sums when you try to define the product of two Laurent series. Infinite sums are problematic when there's no topology around...