In mathematics, given an additive subgroup , the Novikov ring of is the subring of consisting of formal sums such that and . The notion was introduced by S. P. Novikov in the papers that initiated the generalization of Morse theory using a closed one-form instead of a function. The Novikov ring is a principal ideal domain. Let S be the subset of consisting of those with leading term 1. Since the elements of S are unit elements of , the localization of with respect to S is a subring of called the "rational part" of ; it is also a principal ideal domain.