Units in $R[x,x^{-1}]$ Let $R$ be an integral domain. I am looking for a general way to describe the units in 
$$ R[x,x^{-1}] := R[x,y]/(xy-1).$$
Clearly
$$\{rx^n \mid n \in \mathbb{Z}, r \in R^\times \} \subseteq R[x,x^{-1}]^\times$$
is a subgroup, but how do I know whether it's all? I was trying to argue with degrees or to write down multiplication of general elements with Cauchy sums, but it all becomes pretty messy, having also negative exponents.

When $R$ is not integral it's not necessarily true that this is all there is : look at $R= \mathbb Z/4$ and $(2x+1)^2 = 4x^2+4x+1 = 1$ : even in $R[x]$ there can be other units. 
If $R$ is an integral domain, those are indeed the only ones : take $fg = 1$, let $k$ (resp. $j$) be the highest index for which $f_k$ (resp. $g_j$) is nonzero, $k',j'$ similarly but with lowest. 
Then $f_kg_j x^{k+j}$ is the highest degree monomial of $fg$, and $f_kg_j \neq 0$ by integrality. It follows that $k+j = 0$ and $f_kg_j = 1$. 
Similarly, $f_{k'}g_{j'}x^{k'+j'}$ is the lowest degree monomial of $fg$ and $f_{k'}g_{j'} \neq 0$ by integrality. It follows that $k'+j'= 0$. 
But $k'\leq k, j'\leq j$ by definition, therefore since both sums are $0$ we must have $k=k', j=j'$ and in particular, $f$ has only one nonzero monomial (so $f= rx^n$ for some $r,n$), similarly for $g$, and the rest follows. 
