Editing Kernel (algebra) (section)

== Definition ==

=== Group homomorphisms ===
{{Group theory sidebar}}
Let ''G'' and ''H'' be [[group (mathematics)|group]]s and let ''f'' be a [[group homomorphism]] from ''G'' to ''H''. If ''e''<sub>''H''</sub> is the [[identity element]] of ''H'', then the ''kernel'' of ''f'' is the preimage of the singleton set {''e''<sub>''H''</sub>}; that is, the subset of ''G'' consisting of all those elements of ''G'' that are mapped by ''f'' to the element ''e''<sub>''H''</sub>.<ref name="Dummit Group Kernel Definition"/><ref name="Hungerford Kernel">{{harvnb|Hungerford|2014|p=263}}</ref>

The kernel is usually denoted {{nowrap|ker ''f''}} (or a variation).<ref name="Dummit Group Kernel Definition"/> In symbols:
: <math> \ker f = \{g \in G : f(g) = e_{H}\} .</math>

Since a group homomorphism preserves identity elements, the identity element ''e''<sub>''G''</sub> of ''G'' must belong to the kernel.<ref name="Dummit Group Kernel Definition"/> The homomorphism ''f'' is injective if and only if its kernel is only the singleton set {''e''<sub>''G''</sub>}.<ref>{{harvnb|Hungerford|2014|p=264}}</ref>

{{nowrap|ker ''f''}} is a [[subgroup]] of ''G'' and further it is a [[normal subgroup]]. Thus, there is a corresponding [[quotient group]] {{nowrap|''G'' / (ker ''f'')}}. This is isomorphic to ''f''(''G''), the image of ''G'' under ''f'' (which is a subgroup of ''H'' also), by the [[isomorphism theorems|first isomorphism theorem]] for groups.<ref name="Dummit Group FIT"/>

=== Ring homomorphisms ===
{{Ring theory sidebar}}
Let ''R'' and ''S'' be [[ring (mathematics)|ring]]s (assumed [[unital algebra|unital]]) and let ''f'' be a [[ring homomorphism]] from ''R'' to ''S''.
If 0<sub>''S''</sub> is the [[zero element]] of ''S'', then the ''kernel'' of ''f'' is its kernel as additive groups.<ref>{{harvnb|Fraleigh|Katz|2003|p=238}}</ref> It is the preimage of the [[zero ideal]] {{mset|0<sub>''S''</sub>}}, which is, the subset of ''R'' consisting of all those elements of ''R'' that are mapped by ''f'' to the element 0<sub>''S''</sub>.
The kernel is usually denoted {{nowrap|ker ''f''}} (or a variation).
In symbols:
: <math> \operatorname{ker} f = \{r \in R : f(r) = 0_{S}\} .</math>

Since a ring homomorphism preserves zero elements, the zero element 0<sub>''R''</sub> of ''R'' must belong to the kernel.
The homomorphism ''f'' is injective if and only if its kernel is only the singleton set {{mset|0<sub>''R''</sub>}}.
This is always the case if ''R'' is a [[field (mathematics)|field]], and ''S'' is not the [[zero ring]].<ref name="Dummit Ring Kernels and Ideals"/>

Since ker ''f'' contains the multiplicative identity only when ''S'' is the zero ring, it turns out that the kernel is generally not a [[subring]] of ''R.'' The kernel is a sub[[rng (algebra)|rng]], and, more precisely, a two-sided [[ideal (ring theory)|ideal]] of ''R''.
Thus, it makes sense to speak of the [[quotient ring]] {{nowrap|''R'' / (ker ''f'')}}.
The first isomorphism theorem for rings states that this quotient ring is naturally isomorphic to the image of ''f'' (which is a subring of ''S'').<ref name="Dummit Ring Kernels and Ideals"/>

=== Linear maps ===
{{Main|Kernel (linear algebra)}}
Let ''V'' and ''W'' be [[vector space]]s over a [[Field (mathematics)|field]] (or more generally, [[module (mathematics)|modules]] over a [[Ring (mathematics)|ring]]) and let ''T'' be a [[linear map]] from ''V'' to ''W''. If '''0'''<sub>''W''</sub> is the [[zero vector]] of ''W'', then the kernel of ''T'' (or null space<ref name="Axler Kernel Examples"/>) is the [[preimage]] of the [[zero space|zero subspace]] {'''0'''<sub>''W''</sub>}; that is, the [[subset]] of ''V'' consisting of all those elements of ''V'' that are mapped by ''T'' to the element '''0'''<sub>''W''</sub>. The kernel is usually denoted as {{nowrap|ker ''T''}}, or some variation thereof:
: <math> \ker T = \{\mathbf{v} \in V : T(\mathbf{v}) = \mathbf{0}_{W}\} . </math>

Since a linear map preserves zero vectors, the zero vector '''0'''<sub>''V''</sub> of ''V'' must belong to the kernel. The transformation ''T'' is injective if and only if its kernel is reduced to the zero subspace.<ref>{{harvnb|Axler|p=60}}</ref>

The kernel ker ''T'' is always a [[linear subspace]] of ''V''.<ref name="Dummit Dimension">{{harvnb|Dummit|Foote|2004|p=413}}</ref> Thus, it makes sense to speak of the [[quotient space (linear algebra)|quotient space]] {{nowrap|''V'' / (ker ''T'')}}. The first isomorphism theorem for vector spaces states that this quotient space is [[natural isomorphism|naturally isomorphic]] to the [[image (function)|image]] of ''T'' (which is a subspace of ''W''). As a consequence, the [[dimension (linear algebra)|dimension]] of ''V'' equals the dimension of the kernel plus the dimension of the image.<ref name="Dummit Dimension"/>

=== Module homomorphisms ===
Let <math>R</math> be a [[Ring (mathematics)|ring]], and let <math>M</math> and <math>N</math> be <math>R</math>-[[Module (mathematics)|modules]]. If <math>\varphi: M \to N </math> is a module homomorphism, then the kernel is defined to be:<ref name="Dummit Module Kernel Definition">{{harvnb|Dummit|Foote|2004|pp=345–346}}</ref>
: <math> \ker \varphi = \{m \in M \ | \ \varphi (m) = 0\} </math>
Every kernel is a [[submodule]] of the domain module, which means they always contain 0, the additive identity of the module. Kernels of [[abelian group]]s can be considered a particular kind of module kernel when the underlying ring is the [[integer]]s.<ref name="Dummit Module Kernel Definition" />