Engineering / Mathematics — Modules, Vector Spaces and Matrices
Modules: Definitions, Properties and Examples
Modules over a commutative ring, the generalisation of vector spaces, and what changes when scalars need not be invertible.
Executive summary
A module is a vector space over a ring rather than a field. The definition is identical; the theory is substantially different because scalars cannot always be divided by.
Bases need not exist, dimension is not well defined in general, and submodules of free modules need not be free.
Learning objectives
- Define modules and give the standard examples.
- Identify which vector space facts fail over a ring.
- Recognise abelian groups as modules over the integers.
01Definition and examples
Module
An R-module is an abelian group M with a scalar multiplication R × M → M satisfying, for all r, s ∈ R and x, y ∈ M:
r(x + y) = rx + ry, (r + s)x = rx + sx, (rs)x = r(sx), 1x = x.
| Ring R | Module | Interpretation |
|---|---|---|
| A field F | Vector space over F | The familiar case |
| Z | Any abelian group | Scalar action is repeated addition |
| R itself | R as a module over itself | Submodules are exactly the ideals |
| F[X] | Vector space with a linear operator | X acts as the operator |
| Z_n | Abelian group of exponent dividing n | Every element killed by n |
02What fails over a ring
| Vector space fact | Over a general ring |
|---|---|
| Every module has a basis | False — Z_n has no basis over Z |
| Any two bases have equal size | True over commutative rings; false in general |
| Submodules of free modules are free | False over general rings; true over PIDs |
| Every submodule is a direct summand | False — 2Z is not a summand of Z |
| Linearly independent sets extend to bases | False |
The root cause in each case is the same: scalars cannot be divided by. The vector space proofs all normalise a coefficient to 1 at some point, and that step requires invertibility.
Free module and torsion
A module is free if it has a basis. An element x is torsion if rx = 0 for some non-zero r ∈ R.
A free module has no non-zero torsion, so any module with torsion fails to be free.
03Why modules appear here
Linear algebra over finite fields
The Berlekamp algorithm and sparse system solving in index calculus both work in vector spaces over F_p, which are modules where the theory is the familiar one.
Abelian group structure
The structure of Z_n* is a statement about finitely generated Z-modules, and the classification theorem is a module theorem.
Operators as F[X]-modules
Viewing a linear map as an F[X]-module action turns questions about minimal polynomials into module-theoretic ones.
The third of these is used directly in this collection. The minimal polynomial of a linear transformation is the generator of the annihilator ideal of the corresponding F[X]-module, and computing it is what the linearly generated sequence machinery does.
04Frequently asked questions
Why restrict to commutative rings?
Because left and right modules coincide, which removes a layer of bookkeeping, and every application in this collection is over a commutative ring. The non-commutative theory is genuinely different and considerably harder.
Is every module a quotient of a free module?
Yes. Taking one generator per element of the module gives a surjection from a free module, so every module is a quotient of a free one. That is the starting point of homological algebra.
Does dimension make sense for modules?
Only for free modules over commutative rings, where the rank is well defined. For modules with torsion there is no single number playing the role of dimension, which is why the structure theorem gives a list of invariants instead.
Sources and method
Structural reference: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 1, Cambridge University Press, 2005 — book pages 299-301.
This page carries the durable method layer only: definitions, constructions, algorithms, complexity results and selection criteria, authored originally for KEVOS. No text is transcribed or paraphrased from the source, and no numeric tables or benchmark data are reproduced — these are routed to live authoritative sources instead.
Author: Kevin Jogin. Last reviewed 2026-08-07.