Free abelian group

In abstract algebra, a free abelian group or free Z-module is an abelian group with a basis. That is, it is a set together with an associative, commutative, and invertible binary operation, and its basis is a subset of its elements such that every element of the group can be written in one and only one way as a linear combination of basis elements with integer coefficients, finitely many of which are nonzero.  The elements of a free abelian group with basis B are also known as formal sums over B.  Informally, formal sums may also be seen as signed multisets with elements in B. Free abelian groups and formal sums have applications in algebraic topology, where they are used to define chain groups, and in algebraic geometry, where they are used to define divisors.
Every set B has a unique free abelian group with B as its basis. This group may be constructed as a direct sum of copies of the additive group of the integers, with...