Zmodules

Z-modules are abelian groups equipped with a Z-action. For an element a of a Z-module A and an integer n, there is an element n·a in A satisfying n·(a+b) = n·a + n·b, (n+m)·a = n·a + m·a, and 1·a = a. This scalar action endows A with the structure of a module over the ring of integers, and conversely every abelian group can be viewed as a Z-module in this way.

Examples include the integers Z, the finite cyclic groups Z/nZ, and direct sums such as Z^r ⊕ ⊕ Z/d_iZ. The category of Z-modules is equivalent to the category of abelian groups, and module homomorphisms are precisely group homomorphisms. Submodules correspond to subgroups, and the additive structure of abelian groups provides the module operations.

Structure: For finitely generated Z-modules M, there is a canonical decomposition M ≅ Z^r ⊕ T, where T is a finite torsion group. More precisely, the structure theorem for finitely generated modules over a PID states M ≅ Z^r ⊕ ⊕i Z/d_iZ with integers d_i dividing d_{i+1}. The subgroup Tor(M) = {m ∈ M : n·m = 0 for some n ≠ 0} is the torsion part, and r is the rank (the free part).

Applications and notes: This framework yields the classification of finite abelian groups and underpins computations such as the Smith normal form. Divisible Z-modules (for example Q) are injective; unlike vector spaces over a field, Z-modules are not always free or finite-dimensional, reflecting the richer structure of modules over the integers.