Laurent polynomial ring

Definition

Let $R$ be a commutative unital ring. The Laurent polynomial ring over $R$ with indeterminate $x$ is denoted $R[x,x^{-1}]$ and can be defined as follows:

1. It is the ring whose elements are $R$-linear combinations of powers of $x$, where the exponents are allowed to be integers. Addition is coordinate-wise and multiplication is defined $R$-linearly so that on powers of $x$ it is defined by adding the exponents.
2. It is the localization of $R[x]$ at the multiplicatively closed subset of powers of $x$.
3. It is the ring described as $R[x,y]/(xy-1)$.

Particular cases

• If $K$ is a field, the Laurent polynomial ring $K[x,x^{-1}]$ is an intermediate subring between the polynomial ring $K[x]$ and the field of rational functions $K(x)$.