Free abelian group

In abstract algebra, a free abelian group or free Z-module is an abelian group with a basis. Being an abelian group means that it is a set together with an associative, commutative, and invertible binary operation. Conventionally, this operation is thought of as addition and its inverse is thought of as subtraction on the group elements. A basis is a subset of the elements such that every group element can be found by adding or subtracting a finite number of basis elements, and such that, for every group element, its expression as a linear combination of basis elements is unique. For instance, the integers under addition form a free abelian group with basis {1}. Addition of integers is commutative, associative, and has subtraction as its inverse operation, each integer can be formed by using addition or subtraction to combine some number of copies of the number 1, and each integer has a unique representation as an integer multiple of the number 1. Integer lattices also form examples of free abelian groups.