Editing Congruence relation (section)

== Relation with homomorphisms ==
If <math>f:A\, \rightarrow B</math> is a [[homomorphism]] between two algebraic structures (such as [[group homomorphism|homomorphism of groups]], or a [[linear map]] between [[vector space]]s), then the relation <math>R</math> defined by
: <math>a_1\, R\, a_2</math> [[if and only if]] <math>f(a_1) = f(a_2)</math> 
is a congruence relation on <math>A</math>.  By the [[first isomorphism theorem]], the [[image (mathematics)|image]] of ''A'' under <math>f</math> is a substructure of ''B'' [[isomorphism|isomorphic]] to the quotient of ''A'' by this congruence.

On the other hand, the congruence relation <math>R</math> induces a unique homomorphism <math>f: A \rightarrow A/R</math> given by
: <math>f(x) = \{y \mid x \, R \, y\}</math>.

Thus, there is a natural correspondence between the congruences and the homomorphisms of any given algebraic structure.