What are the units in $R[X,X^{-1}]$, where $R$ is a commutative ring with $1$? I know that the question for polynomial rings is a standard textbook exercise. However, I couldn't find a reference for Laurent polynomials, since most people only seem to consider coefficients in an integral domain.



What are the units in the ring of Laurent polynomials?
You can find a more general result in the [paper [1]](http://dx.doi.org/10.1007/s10440-008-9370-8), which determines the units and nilpotents in arbitrary group rings $\rm R[G]$ where $\rm G$ is a unique-product group - which includes ordered groups. As the author remarks, his note was prompted by an earlier [paper [2]](http://homepage.uibk.ac.at/~c70202/jordan/archive/remarks/remarks.pdf) which explicitly treats the Laurent case.


[1](http://dx.doi.org/10.1007/s10440-008-9370-8) Erhard Neher. Invertible and Nilpotent Elements in the Group Algebra of a Unique Product Group  

Acta Appl Math (2009) 108: 135-139  

<http://dx.doi.org/10.1007/s10440-008-9370-8>  

<http://homepage.uibk.ac.at/~c70202/jordan/archive/note/note.pdf>


[2](http://homepage.uibk.ac.at/~c70202/jordan/archive/remarks/remarks.pdf) Ottmar Loos. Remarks on Holger P. Petersson's "Idempotent 2-by-2 matrices"
<http://homepage.uibk.ac.at/~c70202/jordan/archive/remarks/remarks.pdf>


