Zmodule

A Z-module is a module over the ring of integers Z. Concretely, it is an abelian group M equipped with a scalar action of integers, denoted n · x for n ∈ Z and x ∈ M, satisfying the usual module axioms: (m + n) · x = m · x + n · x, m · (x + y) = m · x + m · y, (mn) · x = m · (n · x), and 1 · x = x. In concrete terms, n · x is x added to itself n times (for n > 0), the negative when n < 0, and 0 · x = 0.

Because the action of Z is determined by repeated addition, giving a Z-module structure on M is the same as giving M the structure of an abelian group. Therefore Z-modules are exactly abelian groups, and morphisms of Z-modules are Z-linear maps, i.e., group homomorphisms that respect the Z-action.

A central result for finitely generated Z-modules is the structure theorem for modules over a principal ideal domain: every finitely generated Z-module M is isomorphic to Z^r ⊕ T, where r ≥ 0 is the rank and T is a finite torsion abelian group. The torsion part decomposes further as a direct sum of cyclic groups of prime power order: T ≅ ⊕ Z/p_i^{k_i}. This gives the invariants that classify finitely generated abelian groups.

Submodules of a Z-module correspond to subgroups, and quotients correspond to quotient groups. The category of Z-modules is abelian, supporting kernels, cokernels, exact sequences, and constructions such as Hom and tensor products over Z. Z-modules form a foundational framework in abelian group theory, algebraic topology, and algebraic geometry, where they serve as the basic example of modules and as the target category for many functors.