TITLE: notation for formal Laurent series
QUESTION [7 upvotes]: I've found a few articles that write the ring of formal Laurent series in $t$ as $R((1/t))$, but what's the underlying meaning of $\cdot ((\cdot))$?
A mathematician of my acquaintance swears that $R((t))$, not $R((1/t))$, should be used to denote the ring of formal Laurent series in $t$.  We can't decide who's right without knowing what $\cdot((\cdot))$ means.  (We both agree that $R[[t]]$ denotes the ring of formal power series in $t$ with coefficients in $R$.)

REPLY [7 votes]: I've seen $k((x,y))$ used to mean two different things: the field of fractions of $k[[x,y]]$ and also
$k((x,y)) = \{\sum_{i \geq n} \sum_{j \geq m} f_{i,j}x^iy^j : f_{i,j} \in k; n,m \in \mathbb{Z}\}$
These rings are distinct, e.g. the second one does not contain $\sum_{i \geq 0}y^{-(i+1)}x^i = (y-x)^{-1}$.
In one variable the distinction disappears, and then the only difference between $R((t))$ and $R((1/t))$ is whether the powers of $t$ go to $\infty$ or $-\infty$.