Editing Kernel (algebra) (section)

{{short description|Elements taken to zero by a homomorphism}}
{{Other uses|Kernel (disambiguation){{!}}Kernel}}

[[File:Group homomorphism ver.2.svg|thumb|A [[group homomorphism]] <math>h</math> from the [[group (mathematics)|group]] <math>G</math> to the group <math>H</math> is illustrated, with the groups represented by a blue oval on the left and a yellow circle on the right respectively. The kernel of <math>h</math> is the red circle on the left, as <math>h</math> sends it to the identity element 1 of <math>H</math>.]]

In [[algebra]], the '''kernel''' of a [[homomorphism]] is the relation describing how elements in the [[domain of a function|domain]] of the homomorphism become related in the [[Image (mathematics)|image]].<ref name="McKenzie Kernel">{{harvnb|McKenzie|McNulty|Taylor|1987|pp=27–29}}</ref> A homomorphism is a [[Function (mathematics)|function]] that preserves the underlying [[algebraic structure]] in the domain to its image.

When the algebraic structures involved have an underlying [[Group (mathematics)|group]] structure, the kernel is taken to be the [[preimage]] of the group's identity element in the image, that is, it consists of the elements of the domain mapping to the image's identity.<ref name="Dummit Group Kernel Definition">{{harvnb|Dummit|Foote|2004|p=75}}</ref> For example, the map that sends every [[integer]] to its [[Parity (mathematics)|parity]] (that is, 0 if the number is even, 1 if the number is odd) would be a homomorphism to the integers [[modulo]] 2, and its respective kernel would be the even integers which all have 0 as its parity.<ref name="Dummit Ring Kernel Examples" /> The kernel of a homomorphism of group-like structures will only contain the identity if and only if the homomorphism is [[injective function|injective]], that is if the inverse image of every element consists of a single element. This means that the kernel can be viewed as a measure of the degree to which the homomorphism fails to be injective.<ref name="Dummit Group FIT">{{harvnb|Dummit|Foote|2004|p=97}}</ref>

For some types of structure, such as [[abelian group]]s and [[vector space]]s, the possible kernels are exactly the substructures of the same type. This is not always the case, and some kernels have received a special name, such as [[normal subgroups]] for groups<ref name="Dummit Normal Subgroups">{{harvnb|Dummit|Foote|2004|p=82}}</ref> and [[two-sided ideal]]s for [[ring (mathematics)|rings]].<ref name="Dummit Ring Kernels and Ideals">{{harvnb|Dummit|Foote|2004|pp=239–247}}</ref> The concept of a kernel has been extended to structures such that the inverse image of a single element is not sufficient for deciding whether a homomorphism is injective. In these cases, the kernel is a [[congruence relation]].<ref name="McKenzie Kernel"/>

Kernels allow defining [[quotient object]]s (also called [[quotient (universal algebra)|quotient algebras]] in [[universal algebra]]). For many types of algebraic structure, the [[fundamental theorem on homomorphisms]] (or [[first isomorphism theorem]]) states that [[image (mathematics)|image]] of a homomorphism is [[isomorphism|isomorphic]] to the quotient by the kernel.<ref name="McKenzie Kernel"/><ref name="Dummit Group FIT" />