A Laurent polynomial over may be viewed as a Laurent series in which only finitely many coefficients are non-zero. The ring of Laurent polynomials is an extension of the polynomial ring obtained by "inverting ". More rigorously, it is the localization of the polynomial ring in the multiplicative set consisting of the non-negative powers of . Many properties of the Laurent polynomial ring follow from the general properties of localization. The ring of Laurent polynomials is a subring of the rational functions. The ring of Laurent polynomials over a field is Noetherian (but not Artinian). If is an integral domain, the units of the Laurent polynomial ring have the form , where is a unit of and is an integer. In particular, if is a field then the units of have the form , where is a non-zero element of . The Laurent polynomial ring is isomorphic to the group ring of the group of integers over . More generally, the Laurent polynomial ring in variables is isomorphic to the group ring of the free abelian group of rank . It follows that the Laurent polynomial ring can be endowed with a structure of a commutative, cocommutative Hopf algebra.
A Laurent polynomial over may be viewed as a Laurent series in which only finitely many coefficients are non-zero. The ring of Laurent polynomials is an extension of the polynomial ring obtained by "inverting ". More rigorously, it is the localization of the polynomial ring in the multiplicative set consisting of the non-negative powers of . Many properties of the Laurent polynomial ring follow from the general properties of localization. The ring of Laurent polynomials is a subring of the rational functions. The ring of Laurent polynomials over a field is Noetherian (but not Artinian). If is an integral domain the units of the Laurent polynomial ring have the form , where is a unit of and is an integer. In particular, if is a field then the units of have the form , where is a non-zero element of  The Laurent polynomial ring is isomorphic to the group ring of the group of integers over . More generally, the Laurent polynomial ring in variables is isomorphic to the group ring of the free abelian group of rank . It follows that the Laurent polynomial ring can be endowed with a structure of a commutative, cocommutative Hopf algebra.