zbasis

A zbasis, usually written as Z-basis or Z-basis, refers to a basis for a free abelian group. In this context, a free abelian group is an abelian group that is isomorphic to a finite or countably infinite direct sum of copies of the integers. The concept mirrors that of a basis in vector spaces, but with coefficients restricted to the integers.

Formally, if G is a free abelian group and {b1, ..., bn} is a subset of G, then this set is a Z-basis if every element g in G can be uniquely written as g = z1 b1 + ... + zn bn with zi in Z for all i. The number n is called the rank of G, and G is isomorphic to Z^n under a map sending each basis element to a standard coordinate axis. Conversely, a group that possesses such a basis is called free, and its Z-basis is not unique: any other Z-basis is obtained from the original one by a unimodular change of basis (an element of GL(n, Z)).

Examples include the standard basis e1, ..., en of Z^n, where each ei has a 1 in the i-th position and 0 elsewhere. The group Z^n has many Z-bases, all related by integer invertible matrices. Not every abelian group has a Z-basis; a Z-basis exists precisely for free abelian groups. In the finitely generated case, a group decomposes as a free part Z^r plus a finite torsion part, and only the free part carries a Z-basis. This concept is central in module theory and underpins the classification of finitely generated abelian groups.