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 $$K(x)$$.