The definition, and the one axiom that changes
You already know two examples of a module without knowing the word. A vector space is an abelian group you can scale by elements of a field. An abelian group all by itself is something you can scale by integers — adding an element to itself n times. A module is what you get when you replace “field” or “Z” by an arbitrary ring R: an abelian group M together with a map R × M → M, written (r, m) ↦ rm, satisfying r(m+n)=rm+rn, (r+s)m=rm+sm, (rs)m=r(sm), and 1m=m.
The axioms are letter-for-letter the vector space axioms. Exactly one thing changed: R need not be a field, so a nonzero r need not be invertible. That single relaxation is the whole subject. Over a field you can always divide by a nonzero scalar, which is why every vector space has a basis. Drop division and bases can vanish, dimension can fail to be well-defined, and entirely new phenomena — torsion, non-free modules, projectivity — appear.
A gallery of modules you already know
- Z-modules ARE abelian groups. Given an abelian group A, the action n·a (n copies of a) is the only one possible, so the categories of Z-modules and of abelian groups are literally the same.
- A ring is a module over itself. R acts on R by left multiplication. Its submodules are exactly the ideals of R — this is why ideal theory is module theory in disguise.
- A vector space with an operator is a k[x]-module. Let V be a vector space over a field k and T a linear map. Define x·v = T(v). Then V becomes a module over the polynomial ring k[x], and the structure of V as a k[x]-module encodes the rational and Jordan canonical forms of T.
A submodule N ⊆ M is a subgroup closed under the action (rn ∈ N for all r, n). The intersection of submodules is a submodule, so for any subset S ⊆ M there is a smallest submodule containing S — the submodule generated by S, written RS = { Σ rᵢsᵢ : rᵢ ∈ R, sᵢ ∈ S }. When a single element generates all of M we call M cyclic, the module analogue of a cyclic group.
Torsion and the annihilator: what fields hid from you
Over a field, rv = 0 with r ≠ 0 forces v = 0 — multiply by r⁻¹. Over a general ring this fails, and the failure has a name. An element m is torsion if rm = 0 for some nonzero r ∈ R. The annihilator Ann(m) = { r ∈ R : rm = 0 } is an ideal; Ann(M) = { r : rm = 0 for all m } measures how much of R acts as zero. In a torsion module every element is killed by something nonzero.
Take R = Z and M = Z/6Z, an abelian group hence a Z-module.
The element 2 in Z/6Z is torsion: 3 * 2 = 6 = 0, with 3 != 0 in Z.
So Ann(2) = { n in Z : n*2 = 0 mod 6 } = 3Z.
The element 1 has Ann(1) = 6Z (since 6*1 = 0 is the first hit).
The whole module: Ann(M) = 6Z, because 6 kills everything.
EVERY element of Z/6Z is torsion, so Z/6Z is a torsion Z-module.
Contrast Z itself: n*1 = 0 forces n = 0, so Z is torsion-FREE.
This cannot happen over a field: if k is a field, the only
torsion element of any k-vector space is 0.