Core Structure Theory

Homomorphisms, Kernels and the First Isomorphism Theorem

Homomorphisms as structure-preserving maps, kernels as congruences, and the first isomorphism theorem in the generality where it belongs.

Category Engineering / MathematicsSource II.6Pages 47-51Reading 2 minReviewed 2026-08-07

Learning objectives

Define homomorphism and verify examples
Prove that kernels are congruences and every congruence is a kernel
State and apply the first isomorphism theorem

Homomorphisms

Definition — Homomorphism

A map α: A → B between algebras of the same type is a homomorphism if for every n-ary operation symbol f and all a₁,…,a_n in A: α(f^A(a₁,…,a_n)) = f^B(α(a₁),…,α(a_n)).

For nullary operations the condition reads α(c^A) = c^B: constants map to constants.

Term operations are preserved automatically

By induction on term structure, a homomorphism preserves every term operation, not merely the basic ones. Consequently a homomorphism preserves the truth of every identity: if A satisfies p ≈ q then its homomorphic images do too.

Kernels

Kernels are congruences

For a homomorphism α: A → B, the relation ker(α) = {⟨a, b⟩ : α(a) = α(b)} is a congruence on A.

It is clearly an equivalence relation. For the substitution property, suppose α(a_i) = α(b_i) for each i. Then α(f(a)) = f(α(a)) = f(α(b)) = α(f(b)).

Every congruence is a kernel

For any congruence θ, the natural map ν_θ has kernel θ. So congruences and kernels of homomorphisms are the same thing.

Homomorphism αhas kernel ker(α), a congruence

Congruence θis the kernel of ν_θ

ConclusionCon(A) classifies all homomorphic images of A up to isomorphism

The first isomorphism theorem

First isomorphism theorem

Let α: A → B be a homomorphism. Then A/ker(α) is isomorphic to the image α(A), which is a subalgebra of B. The isomorphism sends a/ker(α) to α(a).

If α is surjective the conclusion reads A/ker(α) ≅ B. Every homomorphism thus factors as a surjection onto a quotient followed by an embedding.

The canonical factorisation

α = ι ∘ β ∘ ν_ker(α), where ν is the natural surjection onto A/ker(α), β is an isomorphism onto the image, and ι is the inclusion of the image into B.

The familiar instances

First isomorphism theorem specialised
Setting	Statement
Groups	G/ker(α) ≅ im(α), with ker(α) the normal subgroup mapping to the identity
Rings	R/I ≅ im(α), with I the ideal mapping to zero
Modules	M/N ≅ im(α)
Vector spaces	V/ker ≅ im — the rank–nullity theorem
Lattices	L/θ ≅ im(α)

One theorem, many corollaries

Each row is normally proved separately in a first course. In universal algebra they are a single theorem instantiated at different types. This is the clearest single demonstration of what the generalisation buys.

Frequently asked questions

Is a bijective homomorphism always an isomorphism?

For algebras, yes — the inverse of a bijective homomorphism is automatically a homomorphism. This differs from topology, where a continuous bijection need not have continuous inverse, and from categories generally.

Why is the image a subalgebra?

Because the image is closed under the operations: applying an operation to images gives the image of the operation applied to preimages, by the homomorphism property.

Source. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition — a corrected re-typesetting of Springer-Verlag Graduate Texts in Mathematics 78 (1981). Section II.6, book pages 47-51.

This page is an original exposition prepared for the KEVOS® knowledge library. It restates and reorganises mathematical results; it is not a reproduction of the source text.