Engineering / Mathematics — Modules, Vector Spaces and Matrices
Module Homomorphisms and Isomorphisms
Module homomorphisms, kernels and images, and the isomorphism theorems in their module form.
Executive summary
A module homomorphism is a map preserving addition and scalar multiplication. Its kernel and image are submodules, and the first isomorphism theorem relates them.
The set of homomorphisms between two modules is itself a module, which is the starting point for homological algebra.
Learning objectives
- Define homomorphisms and verify kernel and image are submodules.
- State the isomorphism theorems.
- Recognise the module structure on the set of homomorphisms.
01Homomorphisms, kernel and image
Module homomorphism
A map f: M → N of R-modules with
f(x + y) = f(x) + f(y) and f(rx) = r f(x).
Over a field this is precisely a linear map.
The kernel {x : f(x) = 0} is a submodule of M, and the image is a submodule of N. Both verifications are immediate from the defining conditions.
A homomorphism is injective exactly when its kernel is zero, by the same argument as for groups: the difference of two elements with equal images lies in the kernel.
02The isomorphism theorems
First isomorphism theorem
For a homomorphism f: M → N,
M / ker f ≅ im f.
Second and third isomorphism theorems
(N₁ + N₂)/N₂ ≅ N₁/(N₁ ∩ N₂).
(M/N₁)/(N₂/N₁) ≅ M/N₂ for N₁ ⊆ N₂ ⊆ M.
These are the same statements as for groups and rings, with the same proofs. The uniformity is not a coincidence — all three are instances of a general categorical pattern, which is where homological algebra begins.
03Hom as a module
The set of homomorphisms from M to N, written Hom_R(M, N), is itself an R-module under pointwise addition and scalar multiplication.
Endomorphisms
Hom(M, M) is a ring under composition — the endomorphism ring. For a vector space it is the matrix ring.
Dual module
Hom(M, R) generalises the dual space. Over a field it has the same dimension; over a general ring it can behave badly.
Exactness failure
Hom does not generally preserve exactness of sequences, and measuring the failure is what the Ext functors do.
The last point connects to the homological algebra collection: the derived functors of Hom are exactly the obstruction to it preserving exact sequences, and that measurement is the content of Ext.
For vector spaces: dim Hom(M, N) = dim M · dim N04Frequently asked questions
Is a bijective homomorphism always an isomorphism?
For modules, yes — the inverse map is automatically a homomorphism. This holds for groups and rings too, and fails in categories such as topological spaces where a continuous bijection need not have a continuous inverse.
Why is Hom(M, N) a module rather than just a group?
Because the ring is commutative: defining (rf)(x) = r f(x) gives a homomorphism only when scalars commute. Over a non-commutative ring Hom is merely an abelian group.
What is the matrix interpretation?
For free modules with chosen bases, a homomorphism is exactly a matrix, and composition is matrix multiplication. The endomorphism ring becomes the ring of square matrices.
Sources and method
Structural reference: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 1, Cambridge University Press, 2005 — book pages 303-306.
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.