Laurent polynomial ring

Summary

The Laurent polynomial ring $R[x, x^{-1}]$ over a ring $R$ consists of finite sums $\sum a_i x^i$ in which the exponents $i$ may be negative. It is the monoid ring of the additive group $\Z$ over $R$. In it the variable $x$ is a unit with inverse $x^{-1}$. It is the group ring of the infinite cyclic group. Over a field it is the coordinate ring of the line with the origin removed.