View source for Kernel (algebra)

{{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" />

== 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" />

== Survey of examples ==

=== Group homomorphisms ===
Let ''G'' be the [[cyclic group]] on 6 elements {{nowrap|{{mset|0, 1, 2, 3, 4, 5}}}} with [[modular arithmetic|modular addition]], ''H'' be the cyclic on 2 elements {{nowrap|{{mset|0, 1}}}} with modular addition, and ''f'' the homomorphism that maps each element ''g'' in ''G'' to the element ''g'' modulo 2 in ''H''. Then {{nowrap|ker ''f'' {{=}} {0, 2, 4} }}, since all these elements are mapped to 0<sub>''H''</sub>. The quotient group {{nowrap|''G'' / (ker ''f'')}} has two elements: {{nowrap|{{mset|0, 2, 4}}}} and {{nowrap|{{mset|1, 3, 5}}}}, and is isomorphic to ''H''.<ref name="Dummit Group Kernel Examples">{{harvnb|Dummit|Foote|2004|pp=78–80}}</ref>

Given a [[isomorphism]] <math>\varphi: G \to H</math>, one has <math>\ker \varphi = 1</math>.<ref name="Dummit Group Kernel Examples" /> On the other hand, if this mapping is merely a homomorphism where ''H'' is the trivial group, then <math>\varphi(g)=1</math> for all <math>g \in G</math>, so thus <math>\ker \varphi = G</math>.<ref name="Dummit Group Kernel Examples" />

Let <math>\varphi: \mathbb{R}^2 \to \mathbb{R}</math> be the map defined as <math>\varphi((x,y)) = x</math>. Then this is a homomorphism with the kernel consisting precisely the points of the form <math>(0,y)</math>. This mapping is considered the "projection onto the x-axis."<ref name="Dummit Group Kernel Examples" /> A similar phenomenon occurs with the mapping <math>f: (\mathbb{R}^\times)^2 \to \mathbb{R}^\times </math> defined as <math>f(a,b)=b</math>, where the kernel is the points of the form <math>(a,1)</math><ref name="Hungerford Kernel"/>

For a non-abelian example, let <math>Q_8</math> denote the [[Quaternion group]], and <math>V_4</math> the [[Klein four-group|Klein 4-group]]. Define a mapping <math>\varphi: Q_8 \to V_4</math> to be:<ref name="Dummit Group Kernel Examples" />
: <math>\varphi(\pm1)=1</math>
: <math>\varphi(\pm i)=a</math>
: <math>\varphi(\pm j)=b</math>
: <math>\varphi(\pm k)=c</math>
Then this mapping is a homomorphism where <math>\ker \varphi = \{ \pm 1 \} </math>.<ref name="Dummit Group Kernel Examples" />

=== Ring homomorphisms ===

Consider the mapping <math> \varphi : \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} </math> where the later ring is the integers modulo 2 and the map sends each number to its [[Parity (mathematics)|parity]]; 0 for even numbers, and 1 for odd numbers. This mapping turns out to be a homomorphism, and since the additive identity of the later ring is 0, the kernel is precisely the even numbers.<ref name="Dummit Ring Kernel Examples">{{harvnb|Dummit|Foote|2004|p=240}}</ref>

Let <math> \varphi: \mathbb{Q}[x] \to \mathbb{Q} </math> be defined as <math>\varphi(p(x))=p(0)</math>. This mapping , which happens to be a homomorphism, sends each polynomial to its constant term. It maps a polynomial to zero [[if and only if]] said polynomial's constant term is 0.<ref name="Dummit Ring Kernel Examples" /> Polynomials with real coefficients can receive a similar homomorphism, with its kernel being the polynomials with constant term 0.<ref>{{harvnb|Hungerford|2014|p=155}}</ref>

=== Linear maps ===
Let <math>\varphi: \mathbb{C}^3 \to \mathbb{C}</math> be defined as <math>\varphi(x,y,z) = x+2y+3z</math>, then the kernel of <math>\varphi</math> (that is, the null space) will be the set of points <math>(x,y,z) \in \mathbb{C}^3</math> such that <math>x+2y+3z=0</math>, and this set is a subspace of <math>\mathbb{C}^3</math> (the same is true for every kernel of a linear map).<ref name="Axler Kernel Examples">{{harvnb|Axler|p=59}}</ref>

If <math>D</math> represents the [[derivative]] operator on real [[polynomial]]s, then the kernel of <math>D</math> will consist of the polynomials with deterivative equal to 0, that is the [[constant function]]s.<ref name="Axler Kernel Examples" />

Consider the mapping <math>(Tp)(x)=x^2p(x)</math>, where <math>p</math> is a polynomial with real coefficients. Then <math>T</math> is a linear map whose kernel is precisely 0, since it is the only polynomial to satisfy <math>x^2p(x) = 0</math> for all <math>x \in \mathbb{R}</math>.<ref name="Axler Kernel Examples" />

== Quotient algebras ==
The kernel of a homomorphism can be used to define a [[Quotient algebra (universal algebra)|quotient algebra]]. For instance, if <math>\varphi: G \to H </math> denotes a group homomorphism, and denote <math>K = \ker \varphi </math>, then consider <math>G/K</math> to be the set of [[Fiber (mathematics)|fibers]] of the homomorphism <math>\varphi</math>, where a fiber is merely the set of points of the domain mapping to a single chosen point in the range.<ref name="Dummit Group Cosets">{{harvnb|Dummit|Foote|2004|pp=74,76–77,80–82}}</ref> If <math>X_a \in G/K</math> denotes the fiber of the element <math> a \in H </math>, then a group operation on the set of fibers can be endowed by <math>X_a X_b = X_{ab}</math>, and <math>G/K</math> is called the quotient group (or factor group), to be read as "G modulo K" or "G mod K".<ref name="Dummit Group Cosets" /> The terminology arises from the fact that the kernel represents the fiber of the identity element of the range, <math>H</math>, and that the remaining elements are simply "translates" of the kernel, so the quotient group is obtained by "dividing out" by the kernel.<ref name="Dummit Group Cosets" />

The fibers can also be described by looking at the domain relative to the kernel; given <math>X \in G/K</math> and any element <math> u \in X </math>, then <math> X = uK = Ku </math> where:<ref name="Dummit Group Cosets" />
: <math> uK = \{ uk \ | \ k \in K \} </math>
: <math> Ku = \{ ku \ | \ k \in K \} </math>
these sets are called the [[coset|left and right cosets]] respectively, and can be defined in general for any arbitrary [[subgroup]] aside from the kernel.<ref name="Dummit Group Cosets" /><ref>{{harvnb|Hungerford|2014|pp=237–239}}</ref><ref>{{harvnb|Fraleigh|Katz|2003|p=97}}</ref> The group operation can then be defined as <math>uK \circ vK = (uk)K</math>, which is well-defined regardless of the choice of representatives of the fibers.<ref name="Dummit Group Cosets" /><ref>{{harvnb|Fraleigh|Katz|2003|p=138}}</ref>

According to the [[Isomorphism theorems|first isomorphism theorem]], there is an isomorphism <math>\mu: G/K \to \varphi(G)</math>, where the later group is the image of the homomorphism <math>\varphi</math>, and the isomorphism is defined as <math>\mu(uK)=\varphi(u)</math>, and such map is also well-defined.<ref name="Dummit Group FIT" /><ref>{{harvnb|Fraleigh|Katz|2003|p=307}}</ref>

For [[Ring (mathematics)|rings]], [[Module (mathematics)|modules]], and [[vector space]]s, one can define the respective quotient algebras via the underlying additive group structure, with cosets represented as <math>x+K</math>. Ring multiplication can be defined on the quotient algebra the same way as in the group (and be well-defined).<ref name="Dummit Ring Kernels and Ideals"/> For a ring <math>R</math> (possibly a [[Field (mathematics)|field]] when describing vector spaces) and a module homomorphism <math>\varphi: M \to N</math> with kernel <math> K = \ker \varphi </math>, one can define scalar multiplication on <math>G/K</math> by <math>r(x+K)=rx+K</math> for <math>r \in R</math> and <math>x \in M</math>, which will also be well-defined.<ref>{{harvnb|Dummit|Foote|2004|pp=345–349}}</ref>

== Kernel structures ==

The structure of kernels allows for the building of quotient algebras from structures satisfying the properties of kernels. Any [[subgroup]] <math>N</math> of a [[Group (mathematics)|group]] <math>G</math> can construct a quotient <math>G/N</math> by the set of all [[coset]]s of <math>N</math> in <math>G</math>.<ref name="Dummit Group Cosets" /> The natural way to turn this into a group, similar to the treatment for the quotient by a kernel, is to define an operation on (left) cosets by <math>uN \cdot vN = (uv)N</math>, however this operation is well defined [[if and only if]] the subgroup <math>N</math> is closed under [[Conjugation (group action)|conjugation]] under <math>G</math>, that is, if <math>g \in G</math> and <math>n \in N</math>, then <math>gng^{-1} \in N</math>. Furthermore, the operation being well defined is sufficient for the quotient to be a group.<ref name="Dummit Group Cosets" /> Subgroups satisfying this property are called [[normal subgroup]]s.<ref name="Dummit Group Cosets" /> Every kernel of a group is a normal subgroup, and for a given normal subgroup <math>N</math> of a group <math>G</math>, the natural projection <math>\pi(g) = gN</math> is a homomorphism with <math>\ker \pi = N</math>, so the normal subgroups are precisely the subgroups which are kernels.<ref name="Dummit Group Cosets" /> The closure under conjugation, however, gives a criterion for when a subgroup is a kernel for some homomorphism.<ref name="Dummit Group Cosets" />

For a [[Ring (mathematics)|ring]] <math>R</math>, treating it as a group, one can take a quotient group via an arbitrary subgroup <math>I</math> of the ring, which will be normal due to the ring's additive group being [[Abelian group|abelian]]. To define multiplication on <math>R/I</math>, the multiplication of cosets, defined as <math>(r+I)(s+I) = rs + I</math> needs to be well-defined. Taking representative <math>r+\alpha</math> and <math>s+\beta</math> of <math>r + I</math> and <math>s + I</math> respectively, for <math>r,s \in R</math> and <math>\alpha, \beta \in I</math>, yields:<ref name="Dummit Ring Kernels and Ideals" />
: <math>(r + \alpha)(s + \beta) + I = rs + I</math>
Setting <math>r = s = 0</math> implies that <math>I</math> is closed under multiplication, while setting <math>\alpha = s = 0</math> shows that <math>r\beta \in I</math>, that is, <math>I</math> is closed under arbitrary multiplication by elements on the left. Similarly, taking <math>r = \beta = 0</math> implies that <math>I</math> is also closed under multiplication by arbitrary elements on the right.<ref name="Dummit Ring Kernels and Ideals" /> Any subgroup of <math>R</math> that is closed under multiplication by any element of the ring is called an [[Ideal (ring theory)|ideal]].<ref name="Dummit Ring Kernels and Ideals" /> Analogously to normal subgroups, the ideals of a ring are precisely the kernels of homomorphisms.<ref name="Dummit Ring Kernels and Ideals" />

== Exact sequence ==
{{Main|Exact sequence}}
[[File:Illustration of an Exact Sequence of Groups.svg|thumb|An exact sequence of groups. At each pair of homomorphism, the image of the previous homomorphism becomes the kernel of the next homomorphism, that is they get sent to the identity element.]]
Kernels are used to define exact sequences of homomorphisms for [[Group (mathematics)|groups]] and [[Module (mathematics)|modules]]. If A, B, and C are modules, then a pair of homomorphisms <math>\psi: A \to B, \varphi: B \to C</math> is said to be exact if <math>\text{image } \psi = \ker \varphi</math>. An exact sequence is then a sequence of modules and homomorphism <math>\cdots \to X_{n-1} \to X_n \to X_{n+1} \to \cdots</math> where each adjacent pair of homomorphisms is exact.<ref>{{harvnb|Dummit|Foote|2004|p=378}}</ref>

== Universal algebra ==
All the above cases may be unified and generalized in [[universal algebra]]. Let ''A'' and ''B'' be [[algebraic structure]]s of a given type and let ''f'' be a homomorphism of that type from ''A'' to ''B''. Then the ''kernel'' of ''f'' is the subset of the [[direct product]] {{nowrap|''A'' × ''A''}} consisting of all those [[ordered pair]]s of elements of ''A'' whose components are both mapped by ''f'' to the same element in ''B''.<ref>{{harvnb|Burris|Sankappanavar|2012|p=44}}</ref><ref name="McKenzie Kernel"/> The kernel is usually denoted {{nowrap|ker ''f''}} (or a variation). In symbols:
: <math>\operatorname{ker} f = \left\{\left(a, b\right) \in A \times A : f(a) = f\left(b\right)\right\}\mbox{.}</math>

The homomorphism ''f'' is injective if and only if its kernel is exactly the diagonal set {{nowrap|{{mset|(''a'', ''a'') : ''a'' &isin; ''A''}}}}, which is always at least contained inside the kernel.<ref>{{harvnb|Burris|Sankappanavar|2012|p=50}}</ref><ref name="McKenzie Kernel"/>

It is easy to see that ker ''f'' is an [[equivalence relation]] on ''A'', and in fact a [[congruence relation]].
Thus, it makes sense to speak of the [[quotient (universal algebra)|quotient algebra]] {{nowrap|''A'' / (ker ''f'')}}.
The [[isomorphism theorem#General|first isomorphism theorem]] in general universal algebra states that this quotient algebra is naturally isomorphic to the image of ''f'' (which is a [[subalgebra]] of ''B'').<ref>{{harvnb|Burris|Sankappanavar|2012|pp=44–46}}</ref>

== See also ==
* [[Kernel (linear algebra)]]
* [[Kernel (category theory)]]
* [[Kernel of a function]]
* [[Equalizer (mathematics)]]
* [[Zero set]]

== Notes ==
{{reflist}}

== References ==
* {{Cite book |last=Axler |first=Sheldon |title=Linear Algebra Done Right |publisher=[[Springer Publishing|Springer]] |edition=4th}}
* {{Cite book |last1=Burris |last2=Sankappanavar |first1=Stanley |first2=H.P. |title=A Course in Universal Algebra |publisher=S. Burris and H.P. Sankappanavar |isbn=978-0-9880552-0-9 |edition=Millennium |publication-date=2012}}
* {{Cite book |last1=Dummit |first1=David Steven |title=Abstract algebra |last2=Foote |first2=Richard M. |date=2004 |publisher=Wiley |isbn=978-0-471-43334-7 |edition=3rd |location=Hoboken, NJ}}
* {{Cite book |last1=Fraleigh |first1=John B. |title=A first course in abstract algebra |last2=Katz |first2=Victor |date=2003 |publisher=Addison-Wesley |isbn=978-0-201-76390-4 |edition=7th |series=World student series |location=Boston}}
* {{Cite book |last=Hungerford |first=Thomas W. |title=Abstract Algebra: an introduction |date=2014 |publisher=Brooks/Cole, Cengage Learning |isbn=978-1-111-56962-4 |edition=3rd |location=Boston, MA}}
* {{Cite book |last1=McKenzie |first1=Ralph |title=Algebras, lattices, varieties |last2=McNulty |first2=George F. |last3=Taylor |first3=W. |date=1987 |publisher=Wadsworth & Brooks/Cole Advanced Books & Software |isbn=978-0-534-07651-1 |series=The Wadsworth & Brooks/Cole mathematics series |location=Monterey, Calif}}

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