Editing Homomorphism (section)

== Kernel ==
{{Main|Kernel (algebra)}}

Any homomorphism <math>f: X \to Y</math> defines an [[equivalence relation]] <math>\sim</math> on <math>X</math> by <math>a \sim b</math> if and only if <math>f(a) = f(b)</math>. The relation <math>\sim</math> is called the '''kernel''' of <math>f</math>. It is a [[congruence relation]] on <math>X</math>. The [[quotient set]] <math>X/{\sim}</math> can then be given a structure of the same type as <math>X</math>, in a natural way, by defining the operations of the quotient set by <math>[x] \ast [y] =  [x \ast y]</math>, for each operation <math>\ast</math> of <math>X</math>. In that case the image of <math>X</math> in <math>Y</math> under the homomorphism <math>f</math> is necessarily [[isomorphic]] to <math>X/\!\sim</math>; this fact is one of the [[isomorphism theorem]]s.

When the algebraic structure is a  [[group (mathematics)|group]] for some operation, the [[equivalence class]] <math>K</math> of the [[identity element]] of this operation suffices to characterize the equivalence relation. In this case, the quotient by the equivalence relation is denoted by <math>X/K</math> (usually read as "<math>X</math> [[Ideal (ring theory)|mod]] <math>K</math>"). Also in this case, it is <math>K</math>, rather than <math>\sim</math>, that is called the [[kernel (algebra)|kernel]] of <math>f</math>. The kernels of homomorphisms of a given type of algebraic structure are naturally equipped with some structure. This structure type of the kernels is the same as the considered structure, in the case of [[abelian group]]s, [[vector space]]s and [[module (mathematics)|modules]], but is different and has received a specific name in other cases, such as [[normal subgroup]] for kernels of [[group homomorphisms]] and [[ideal (ring theory)|ideals]] for kernels of [[ring homomorphism]]s (in the case of non-commutative rings, the kernels are the [[two-sided ideal]]s).