Modules are vector spaces that forgot their scalars were a field
Replace the field of scalars by a ring and you get a module: still an abelian group with distributive scaling, but bases may vanish and ranks may misbehave. Every abelian group is secretly a module over the integers.
Scalar multiplication must respect ring multiplication and distribute over addition in both the ring and the module. Modules sit at the center of commutative algebra, homological algebra, algebraic geometry, and algebraic topology, and they intertwine with group representation theory. Ideals and quotient rings become modules, so arguments unify. Over noncommutative rings one must distinguish left and right modules and left ideals carefully.
Much of the theory tries to recover vector-space comforts over well-behaved rings such as principal ideal domains. Yet not every module has a basis; free modules that do may lack a unique rank if the ring fails the invariant basis number property. (Bases for general vector spaces need the axiom of choice.) A left R-module pairs an abelian group with a scalar action from R × M; right modules reverse the side of the associativity law. When R is commutative the two notions coincide. Over a field, modules are exactly vector spaces. A K[x]-module is a vector space plus a commuting linear operator—hence rational and Jordan forms via the structure theorem for finitely generated modules over a PID.
Abelian groups are Z-modules in a unique way via repeated addition; torsion groups need not have bases. Decimal fractions form a Z-module with no basis in the linear-algebra sense yet torsion-free rank one. Free modules Rn illustrate finite rank; matrix rings Mn(R) act so that categories of R-modules and Mn(R)-modules are equivalent. Bimodules carry compatible left and right actions. Wherever linear algebra's proofs used division by scalars, module theory asks which theorems survive when only a ring remains.
Source: Module (mathematics)
