for each function) and the free abelian group over the complex numbers. The rational functions that have a nonzero limiting value at infinity (the meromorphic functions on the Riemann sphere) form a subgroup of this group in which the sum of the multiplicities is zero. This construction has been generalized, in algebraic geometry, to the notion of a divisor. There are different definitions of divisors, but in general they form an abstraction of a codimension-one subvariety of an algebraic variety, the set of solution points of a system of polynomial equations. In the case where the system of equations has one degree of freedom (its solutions form an algebraic curve or Riemann surface), a subvariety has codimension one when it consists of isolated points, and in this case a divisor is again a signed multiset of points from the variety. The meromorphic functions on a compact Riemann surface have finitely many zeros and poles, and their divisors form a subgroup of a free abelian group over the points of the surface, with multiplication or division of functions corresponding to addition or subtraction of group elements. To be a divisor, an element of the free abelian group must have multiplicities summing to zero, and meet certain additional constraints depending on the surface. Group rings The integral group ring , for any group , is a ring whose additive group is the free abelian group over . When is finite and abelian, the multiplicative group of units in has the structure of a direct product of a finite group and a finitely generated free abelian group. References Abelian group theory Properties of groups Free algebraic structures