Editing Kernel (algebra)

{{short description|Inverse image of zero under a homomorphism}}
{{more citations needed|date=April 2025}}
[[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]] (function that preserves the [[algebraic structure|structure]]) is generally the [[inverse image]] of 0 (except for [[group (mathematics)|groups]] whose operation is denoted multiplicatively, where the kernel is the inverse image of 1). An important special case is the [[kernel (linear algebra)|kernel of a linear map]]. The [[kernel (matrix)|kernel of a matrix]], also called the ''null space'', is the kernel of the linear map defined by the matrix. 

The kernel of a homomorphism is reduced to 0 (or 1) 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.  

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, sometimes, the possible kernels have received a special name, such as [[normal subgroup]] for groups and [[two-sided ideal]]s for [[ring (mathematics)|rings]].

Kernels allow defining [[quotient object]]s (also called [[quotient (universal algebra)|quotient algebras]] in [[universal algebra]], and [[cokernel]]s in [[category theory]]). 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.

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]].

This article is a survey for some important types of kernels in algebraic structures.

== History ==
The mathematician [[Lev Pontryagin|Pontryagin]] is credited with using the word "kernel" in 1931 to describe the elements of a group that were sent to the identity element in another group. <ref>{{cite journal |last1=Pontrjagin |first1=L. |title=Über den algebraischen Inhalt topologischer Dualitätssätze |journal=Mathematische Annalen |date=1931 |volume=105 |page=186 |doi=10.1007/BF01455814 }} Cited in {{cite web |last1=Conrad |first1=Keith |title=Homomorphisms |url=https://kconrad.math.uconn.edu/blurbs/grouptheory/homomorphisms.pdf |website=Expository papers |access-date=15 April 2025}}</ref>

== 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=":0">{{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}}</ref><ref name=":1">{{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}}</ref><ref name=":2">{{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}}</ref>

The kernel is usually denoted {{nowrap|ker ''f''}} (or a variation). 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.

The homomorphism ''f'' is injective if and only if its kernel is only the singleton set {''e''<sub>''G''</sub>}. If ''f'' were not injective, then the non-injective elements can form a distinct element of its kernel: there would exist {{nowrap|''a'', ''b'' &isin; ''G''}} such that {{nowrap|''a'' ≠ ''b''}} and {{nowrap|1=''f''(''a'') = ''f''(''b'')}}. Thus {{nowrap|1=''f''(''a'')''f''(''b'')<sup>−1</sup> = ''e''<sub>''H''</sub>}}. ''f'' is a group homomorphism, so inverses and group operations are preserved, giving {{nowrap|1=''f''(''ab''<sup>−1</sup>) = ''e''<sub>''H''</sub>}}; in other words, {{nowrap|''ab''<sup>−1</sup> &isin; ker ''f''}}, and ker ''f'' would not be the singleton. Conversely, distinct elements of the kernel violate injectivity directly: if there would exist an element {{nowrap|''g'' ≠ ''e''<sub>''G''</sub> &isin; ker ''f''}}, then {{nowrap|1=''f''(''g'') = ''f''(''e''<sub>''G''</sub>) = ''e''<sub>''H''</sub>}}, thus ''f'' would not be injective.

{{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.

=== 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 name=":1" /> 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]].

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''). (Note that rings need not be unital for the kernel definition).

=== 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>{{Cite book |last=Axler |first=Sheldon |title=Linear Algebra Done Right |publisher=[[Springer Publishing|Springer]] |edition=4th}}</ref><ref name=":0" />) 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.

The kernel ker ''T'' is always a [[linear subspace]] of ''V''.<ref name=":0" /> 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.

One can define kernels for homomorphisms between modules over a [[ring (mathematics)|ring]] in an analogous manner. This includes kernels for homomorphisms between [[abelian group]]s as a special case.<ref name=":0" /> This example captures the essence of kernels in general [[abelian categories]]; see [[Kernel (category theory)]].

=== 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=":0" />
: <math> \ker \varphi = \{m \in M \ | \ \varphi (m) = 0\} </math>
Every kernel is a [[submodule]] of the domain module.<ref name=":0" />

=== Monoid homomorphisms ===
{{unreferenced section|date=April 2025}}
Let ''M'' and ''N'' be [[monoid (algebra)|monoid]]s and let ''f'' be a [[monoid homomorphism]] from ''M'' to ''N''. Then the ''kernel'' of ''f'' is the subset of the [[direct product]] {{nowrap|''M'' × ''M''}} consisting of all those [[ordered pair]]s of elements of ''M'' whose components are both mapped by ''f'' to the same element in ''N''{{citation needed|date=April 2025}}. The kernel is usually denoted {{nowrap|ker ''f''}} (or a variation thereof). In symbols:
: <math>\operatorname{ker} f = \left\{\left(m, m'\right) \in M \times M : f(m) = f\left(m'\right)\right\}.</math>

Since ''f'' is a [[function (mathematics)|function]], the elements of the form {{nowrap|(''m'', ''m'')}} must belong to the kernel. The homomorphism ''f'' is injective if and only if its kernel is only the [[Equality (mathematics)|diagonal set]] {{nowrap|{{mset|(''m'', ''m'') : ''m'' in ''M''}}}}.

It turns out that {{nowrap|ker ''f''}} is an [[equivalence relation]] on ''M'', and in fact a [[congruence relation]]. Thus, it makes sense to speak of the [[quotient monoid]] {{nowrap|''M'' / (ker ''f'')}}. The first isomorphism theorem for monoids states that this quotient monoid is naturally isomorphic to the image of ''f'' (which is a [[submonoid]] of ''N''; for the congruence relation).

This is very different in flavor from the above examples. In particular, the preimage of the identity element of ''N'' is ''not'' enough to determine the kernel of ''f''.

== 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}}}}. It is indeed isomorphic to ''H''.

Given a [[isomorphism]] <math>\varphi: G \to H</math>, one has <math>\ker \varphi = 1</math>.<ref name=":0" /> 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=":0" /> 

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=":0" /> 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=":2" />

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:
: <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=":0" />

=== 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=":0" />

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=":0" /> If we instead work with polynomials with real coefficients, then we again receive a homomorphism with its kernel being the polynomials with constant term 0.<ref name=":2" />

=== Linear maps ===
{{unreferenced section|date=April 2025}}
If ''V'' and ''W'' are [[finite-dimensional vector space|finite-dimensional]] and [[basis (linear algebra)|bases]] have been chosen, then ''T'' can be described by a [[matrix (mathematics)|matrix]] ''M'', and the kernel can be computed by solving the homogeneous [[system of linear equations]] {{nowrap|1=''M'''''v''' = '''0'''}}. In this case, the kernel of ''T'' may be identified to the [[kernel (matrix)|kernel of the matrix]] ''M'', also called "null space" of ''M''. The dimension of the null space, called the nullity of ''M'', is given by the number of columns of ''M'' minus the [[rank (matrix theory)|rank]] of ''M'', as a consequence of the [[rank–nullity theorem]].

Solving [[homogeneous differential equation]]s often amounts to computing the kernel of certain [[differential operator]]s.
For instance, in order to find all twice-[[differentiable function]]s ''f'' from the [[real line]] to itself such that
: <math>x f''(x) + 3 f'(x) = f(x),</math>
let ''V'' be the space of all twice differentiable functions, let ''W'' be the space of all functions, and define a linear operator ''T'' from ''V'' to ''W'' by
: <math>(Tf)(x) = x f''(x) + 3 f'(x) - f(x)</math>
for ''f'' in ''V'' and ''x'' an arbitrary [[real number]].
Then all solutions to the differential equation are in {{nowrap|ker ''T''}}.

== 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 we set <math>K = \ker \varphi </math>, we can 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=":0" /> If <math>X_a \in G/K</math> denotes the fiber of the element <math> a \in H </math>, then we can give a group operation on the set of fibers by <math>X_a X_b = X_{ab}</math>, and we call <math>G/K</math> the quotient group (or factor group), to be read as "G modulo K" or "G mod K".<ref name=":0" /> 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=":0" />

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=":0" />
: <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=":0" /><ref name=":2" /><ref name=":1" /> 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=":0" /><ref name=":1" />

According to the [[Isomorphism theorems|first isomorphism theorem]], we have 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=":0" /><ref name=":1" />

For [[Ring (mathematics)|rings]], [[Module (mathematics)|modules]], and [[vector space|vector spaces]], 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). 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 name=":0" />

== 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|cosets]] of <math>N</math> in <math>G</math>.<ref name=":0" /> 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|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=":0" /> Subgroups satisfying this property are called [[Normal subgroup|normal subgroups]].<ref name=":0" /> 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=":0" /> The closure under conjugation, however, gives an "internal"<ref name=":0" /> criterion for when a subgroup is a kernel for some homomorphism.

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=":0" />
: <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=":0" /> 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=":0" /> Analogously to normal subgroups, the ideals of a ring are precisely the kernels of homomorphisms.<ref name=":0" />

== 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 name=":0" />

== 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 name=":3">{{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}}</ref><ref name=":4">{{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}}</ref> 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 name=":3" /><ref name=":4" />

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 name=":3" /><ref name=":4" />

Note that the definition of kernel here (as in the monoid example) doesn't depend on the algebraic structure; it is a purely [[set (mathematics)|set]]-theoretic concept.
For more on this general concept, outside of abstract algebra, see [[kernel of a function]].

== Algebras with nonalgebraic structure ==
{{unreferenced section|date=April 2025}}
Sometimes algebras are equipped with a nonalgebraic structure in addition to their algebraic operations.
For example, one may consider [[topological group]]s or [[topological vector space]]s, which are equipped with a [[topology (structure)|topology]].
In this case, we would expect the homomorphism ''f'' to preserve this additional structure{{citation needed|date=April 2025}}; in the topological examples, we would want ''f'' to be a [[continuous map]].
The process may run into a snag with the quotient algebras, which may not be well-behaved.
In the topological examples, we can avoid problems by requiring that topological algebraic structures be [[Hausdorff space|Hausdorff]] (as is usually done); then the kernel (however it is constructed) will be a [[closed set]] and the [[quotient space (topology)|quotient space]] will work fine (and also be Hausdorff){{citation needed|date=April 2025}}.

== Kernels in category theory ==
{{unreferenced section|date=April 2025}}
The notion of ''kernel'' in [[category theory]] is a generalization of the kernels of abelian algebras; see [[Kernel (category theory)]].
The categorical generalization of the kernel as a congruence relation is the ''[[kernel pair]]''.
(There is also the notion of [[difference kernel]], or binary [[equalizer (mathematics)|equalizer]].)

== See also ==
* [[Kernel (linear algebra)]]
* [[Zero set]]

== References ==
{{reflist}}
{{refbegin}}
{{cite book
 | last=Lang | first=Serge | author-link=Serge Lang
 | year=2002
 | title=Algebra
 | publisher=[[Springer Science+Business Media|Springer]]
 | series=[[Graduate Texts in Mathematics]]
 | isbn=0-387-95385-X
}}
{{refend}}

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[[Category:Isomorphism theorems]]
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