General algebraic systems Congruences and quotients Concept
Congruence relation
Summary
A congruence relation on an algebra is an equivalence relation that is compatible with all the operations. Replacing elements by congruent ones never changes the class of the result. Congruences are exactly the relations that allow a quotient algebra, and they are exactly the kernels of homomorphisms. In groups they correspond to normal subgroups, and in rings to two-sided ideals.
Where this leads
19 pages build on this one.
- Quotient algebra
- Factor congruence
- Essential extension
- Variety as a reflective subcategory
- Fully invariant congruence
- Congruence generated by a relation
- Simple algebra
- Reduced product
- Birkhoff's subdirect representation theorem
- Mal'cev's theorem on congruence permutability
- Polynomial operation
- Correspondence theorem for congruences
- Homomorphism theorem
- Permuting congruences
- Congruence-permutable variety
- Subdirectly irreducible algebra
- Commutator theory
- Associative quotient of a magma
- Congruence lattice
Further reading
About this page
| Tag | DCSXXT4SER |
|---|---|
| URI | m2://DCSXXT4SER |
| Kind | concept |
| Writing | W0 stub |
| MSC | 08A30 |
| Specializes | Equivalence relation |
| Same as | |
| Changed | 2 November 2026 |
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Used by
- Quotient algebra
- Factor congruence
- Essential extension
- Variety as a reflective subcategory
- Fully invariant congruence
- Congruence generated by a relation
- Simple algebra
- Reduced product
- Birkhoff's subdirect representation theorem
- Mal'cev's theorem on congruence permutability
- Polynomial operation
- Correspondence theorem for congruences
- Homomorphism theorem
- Permuting congruences
- Congruence-permutable variety
- Subdirectly irreducible algebra
- Commutator theory
- Associative quotient of a magma
- Congruence lattice