Engineering / Mathematics — Modules, Vector Spaces and Matrices
Submodules and Quotient Modules
Submodules, quotient modules, and the correspondence between submodules of a quotient and those of the original.
Executive summary
A submodule is a subset closed under the operations; a quotient module is formed by collapsing a submodule to zero. Both constructions mirror the group and ring cases exactly.
The correspondence theorem relates submodules of a quotient to those of the original containing the collapsed one.
Learning objectives
- Define submodules and verify the quotient construction.
- State the correspondence theorem.
- Recognise ideals as submodules of the ring.
01Submodules
Submodule
A subset N ⊆ M is a submodule if it is a subgroup under addition and closed under scalar multiplication: rx ∈ N for all r ∈ R, x ∈ N.
Viewing R as a module over itself, the submodules are exactly the ideals. This is why ideal theory and module theory develop in parallel, and why results about one often transfer to the other.
- Intersections of submodules are submodules; unions generally are not.
- The sum
N₁ + N₂ = {x + y}is the smallest submodule containing both. - The submodule generated by a set is the set of finite R-linear combinations of its elements.
- A module is finitely generated if some finite set generates it.
02Quotient modules
Quotient module
For a submodule N ⊆ M, the quotient M/N is the set of cosets x + N with operations
(x + N) + (y + N) = (x + y) + N and r(x + N) = rx + N.
Well-definedness of the scalar action requires N to be closed under scalar multiplication, which is exactly the submodule condition. Unlike groups, no normality hypothesis is needed — every submodule of a module over a commutative ring gives a quotient.
03The correspondence theorem
Correspondence
For a submodule N ⊆ M, the submodules of M/N correspond bijectively to the submodules of M containing N, via P ↦ P/N.
The correspondence preserves inclusion, sums and intersections.
| Structure | Sub-object | Quotient condition |
|---|---|---|
| Group | Subgroup | Normality required |
| Ring | Ideal | Two-sided ideal required |
| Module | Submodule | No extra condition |
| Vector space | Subspace | No extra condition |
The theorem is used constantly as a bookkeeping device: it converts questions about a quotient into questions about the original module, where more structure is available.
04Frequently asked questions
Why do modules need no normality condition?
Because the underlying group is abelian, so every subgroup is normal. The scalar closure condition is the only additional requirement, and it is built into the definition of a submodule.
Is a submodule of a finitely generated module finitely generated?
Not in general. It is true over Noetherian rings, which includes fields, the integers and polynomial rings over them — so it holds in every case arising in this collection.
What is the analogue of a normal subgroup here?
There is none needed. The full analogy is: normal subgroups of groups, two-sided ideals of rings, and all submodules of modules — each being exactly the sub-objects by which one can quotient.
Sources and method
Structural reference: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 1, Cambridge University Press, 2005 — book pages 301-303.
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.