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| In the various branches of [[mathematics]] that fall under the heading of [[abstract algebra]], the '''kernel''' of a [[homomorphism]] measures the degree to which the homomorphism fails to be [[injective function|injective]].<ref>{{cite book | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract Algebra | publisher=[[John Wiley & Sons]] | year=2004 | edition=3rd | isbn=0-471-43334-9}}</ref><ref>{{cite book | last=Lang | first=Serge | authorlink=Serge Lang | title=Algebra | publisher=[[Springer Science+Business Media|Springer]] | series=[[Graduate Texts in Mathematics]] | year=2002 | isbn=0-387-95385-X}}</ref> 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.
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| The definition of kernel takes various forms in various contexts. But in all of them, the kernel of a homomorphism is trivial (in a sense relevant to that context) if and only if the homomorphism is [[injective function|injective]]. The [[fundamental theorem on homomorphisms]] (or [[first isomorphism theorem]]) is a theorem, again taking various forms, that applies to the [[quotient algebra]] defined by the kernel.
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| In this article, we first survey kernels for some important types of [[algebraic structure]]s; then we give general definitions from [[universal algebra]] for generic algebraic structures.
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| ==Survey of examples==
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| === Linear maps ===
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| {{Main|Kernel (linear algebra)}}
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| Let ''V'' and ''W'' be [[vector space]]s (or more generally [[module (mathematics)|modules]]) 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'' 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 "ker ''T''", or some variation thereof:
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| :<math> \operatorname{ker} T := \{\mathbf{v} \in V : T\mathbf{v} = \mathbf{0}_{W}\}\text{.} </math>
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| 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.
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| The kernel ker ''T'' is always a [[linear subspace]] of ''V''. Thus, it makes sense to speak of the [[quotient space (linear algebra)|quotient space]] ''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.
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| 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]] ''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]].
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| Solving [[homogeneous differential equation]]s often amounts to computing the kernel of certain [[differential operator]]s.
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| For instance, in order to find all twice-[[differentiable function]]s ''f'' from the [[real line]] to itself such that
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| : ''x'' ''f''′′(''x'') + 3''f''′(''x'') = ''f''(''x''),
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| 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
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| : (''Tf'')(''x'') = ''x'' ''f''′′(''x'') + 3''f''′(''x'') − ''f''(''x'')
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| for ''f'' in ''V'' and ''x'' an arbitrary [[real number]].
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| Then all solutions to the differential equation are in ker ''T''.
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| One can define kernels for [[homomorphism]]s between [[module (mathematics)|module]]s over a [[ring (mathematics)|ring]] in an analogous manner. This includes kernels for homomorphisms between [[abelian group]]s as a special case. This example captures the essence of kernels in general [[abelian categories]]; see [[Kernel (category theory)]].
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| ===Group homomorphisms===
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| Let ''G'' and ''H'' be [[group (mathematics)|group]]s and let ''f'' be a [[group homomorphism]] from ''G'' to ''H''.
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| 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>.
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| The kernel is usually denoted "ker ''f''" (or a variation).
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| In symbols:
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| : <math> \operatorname{ker} f := \{g \in G : f(g) = e_{H}\}\mbox{.}</math> | |
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| Since a group homomorphism preserves identity elements, the identity element ''e''<sub>''G''</sub> of ''G'' must belong to the kernel.
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| The homomorphism ''f'' is injective if and only if its kernel is only the singleton set {''e''<sub>''G''</sub>}.
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| It turns out that ker ''f'' is not only a [[subgroup]] of ''G'' but in fact a [[normal subgroup]].
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| Thus, it makes sense to speak of the [[quotient group]] ''G''/(ker ''f'').
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| The first isomorphism theorem for groups states that this quotient group is [[natural isomorphism|naturally isomorphic]] to the [[image (function)|image]] of ''f'' (which is a subgroup of ''H'').
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| In the special case of [[abelian group]]s, this works in exactly the same way as in the previous section.
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| ===Ring homomorphisms===
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| Let ''R'' and ''S'' be [[ring (mathematics)|ring]]s (assumed [[unital algebra|unital]]) and let ''f'' be a [[ring homomorphism]] from ''R'' to ''S''.
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| If 0<sub>''S''</sub> is the [[zero element]] of ''S'', then the ''kernel'' of ''f'' is its kernel as linear map over the integers, or, equivalently, as additive groups. It is the preimage of the [[zero ideal]] {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>.
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| The kernel is usually denoted "ker ''f''" (or a variation).
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| In symbols:
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| : <math> \operatorname{ker} f := \{r \in R : f(r) = 0_{S}\}\mbox{.} \! </math>
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| Since a ring homomorphism preserves zero elements, the zero element 0<sub>''R''</sub> of ''R'' must belong to the kernel.
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| The homomorphism ''f'' is injective if and only if its kernel is only the singleton set {0<sub>''R''</sub>}.
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| It turns out that, although ker ''f'' is generally not a [[subring]] of ''R'' since it may not contain the multiplicative identity if ''S'' is not the [[null ring]] (although the kernel is a subring for nonunital rings). Nevertheless it is a two-sided [[ideal (ring theory)|ideal]] of ''R''.
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| Thus, it makes sense to speak of the [[quotient ring]] ''R''/(ker ''f'').
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| The first isomorphism theorem for rings states that this quotient ring is [[natural isomorphism|naturally isomorphic]] to the [[image (function)|image]] of ''f'' (which is a subring of ''S''). (note that rings need not be unital for the kernel definition).
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| To some extent, this can be thought of as a special case of the situation for modules, since these are all [[bimodule]]s over a ring ''R'':
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| * ''R'' itself;
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| * any two-sided ideal of ''R'' (such as ker ''f'');
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| * any quotient ring of ''R'' (such as ''R''/(ker ''f'')); and
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| * the [[codomain]] of any ring homomorphism whose domain is ''R'' (such as ''S'', the codomain of ''f'').
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| However, the isomorphism theorem gives a stronger result, because ring isomorphisms preserve multiplication while module isomorphisms (even between rings) in general do not.
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| This example captures the essence of kernels in general [[Mal'cev algebra]]s.
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| ===Monoid homomorphisms===
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| Let ''M'' and ''N'' be [[monoid (algebra)|monoid]]s and let ''f'' be a [[monoid homomorphism]] from ''M'' to ''N''.
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| Then the ''kernel'' of ''f'' is the subset of the [[direct product]] ''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''.
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| The kernel is usually denoted "ker ''f''" (or a variation).
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| In symbols:
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| : <math> \operatorname{ker} f := \{(m,m') \in M \times M : f(m) = f(m')\}\mbox{.} \! </math>
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| Since ''f'' is a [[function (mathematics)|function]], the elements of the form (''m'',''m'') must belong to the kernel.
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| The homomorphism ''f'' is injective if and only if its kernel is only the [[Equality (mathematics)|diagonal set]] {(m,m) : ''m'' in ''M''}.
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| It turns out that ker ''f'' is an [[equivalence relation]] on ''M'', and in fact a [[congruence relation]].
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| Thus, it makes sense to speak of the [[quotient monoid]] ''M''/(ker ''f'').
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| The first isomorphism theorem for monoids states that this quotient monoid is [[natural isomorphism|naturally isomorphic]] to the [[image (function)|image]] of ''f'' (which is a [[submonoid]] of ''N''),(for the congruence relation).
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| This is very different in flavour from the above examples.
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| In particular, the preimage of the identity element of ''N'' is ''not'' enough to determine the kernel of ''f''.
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| This is because monoids are not Mal'cev algebras.
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| ==Universal algebra==
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| All the above cases may be unified and generalized in [[universal algebra]].
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| ===General case===
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| 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''.
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| Then the ''kernel'' of ''f'' is the subset of the [[direct product]] ''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''.
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| The kernel is usually denoted "ker ''f''" (or a variation).
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| In symbols:
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| : <math> \operatorname{ker} f := \{(a,a') \in A \times A : f(a) = f(a')\}\mbox{.} \! </math>
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| Since ''f'' is a [[function (mathematics)|function]], the elements of the form (''a'',''a'') must belong to the kernel.
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| The homomorphism ''f'' is injective if and only if its kernel is only the diagonal set {(a,a) : ''a'' in ''A''}.
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| It turns out that ker ''f'' is an [[equivalence relation]] on ''A'', and in fact a [[congruence relation]].
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| Thus, it makes sense to speak of the [[quotient algebra]] ''A''/(ker ''f'').
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| The first isomorphism theorem in general universal algebra states that this quotient algebra is [[natural isomorphism|naturally isomorphic]] to the [[image (function)|image]] of ''f'' (which is a [[subalgebra]] of ''B'').
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| 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.
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| For more on this general concept, outside of abstract algebra, see [[kernel of a function]].
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| ===Mal'cev algebras===
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| {{Main|Malcev algebra}}
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| In the case of Mal'cev algebras, this construction can be simplified. Every Mal'cev algebra has a special [[neutral element]] (the [[null vector]] in the case of [[vector space]]s, the [[identity element]] in the case of [[commutative group]]s, and the [[zero element]] in the case of [[ring (mathematics)|ring]]s or [[module (mathematics)|module]]s). The characteristic feature of a Mal'cev algebra is that we can recover the entire equivalence relation ker ''f'' from the [[equivalence class]] of the neutral element.
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| To be specific, let ''A'' and ''B'' be Mal'cev algebraic structures of a given type and let ''f'' be a homomorphism of that type from ''A'' to ''B''. If ''e''<sub>''B''</sub> is the neutral element of ''B'', then the ''kernel'' of ''f'' is the [[preimage]] of the [[singleton set]] {''e''<sub>''B''</sub>}; that is, the [[subset]] of ''A'' consisting of all those elements of ''A'' that are mapped by ''f'' to the element ''e''<sub>''B''</sub>.
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| The kernel is usually denoted "ker ''f''" (or a variation). In symbols:
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| : <math> \mathop{\mathrm{ker}} f := \{a \in A : f(a) = e_{B}\}\mbox{.} \! </math>
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| Since a Mal'cev algebra homomorphism preserves neutral elements, the identity element ''e''<sub>''A''</sub> of ''A'' must belong to the kernel. The homomorphism ''f'' is injective if and only if its kernel is only the singleton set {''e''<sub>''A''</sub>}.
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| The notion of [[ideal (ring theory)|ideal]] generalises to any Mal'cev algebra (as [[linear subspace]] in the case of vector spaces, [[normal subgroup]] in the case of groups, two-sided ideals in the case of rings, and [[submodule]] in the case of [[module (algebra)|module]]s).
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| It turns out that ker ''f'' is not [[subalgebra]] of ''A'', but it is an ideal.
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| Then it makes sense to speak of the [[quotient algebra]] ''G''/(ker ''f'').
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| The first isomorphism theorem for Mal'cev algebras states that this quotient algebra is naturally isomorphic to the image of ''f'' (which is a subalgebra of ''B'').
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| The connection between this and the congruence relation is for more general types of algebras is as follows.
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| First, the kernel-as-an-ideal is the equivalence class of the neutral element ''e''<sub>''A''</sub> under the kernel-as-a-congruence. For the converse direction, we need the notion of [[quotient]] in the Mal'cev algebra (which is [[division (mathematics)|division]] on either side for groups and [[subtraction]] for vector spaces, modules, and rings).
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| Using this, elements ''a'' and ''b'' of ''A'' are equivalent under the kernel-as-a-congruence if and only if their quotient ''a''/''b'' is an element of the kernel-as-an-ideal.
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| ==Algebras with nonalgebraic structure==
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| Sometimes algebras are equipped with a nonalgebraic structure in addition to their algebraic operations.
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| For example, one may consider [[topological group]]s or [[topological vector space]]s, with are equipped with a [[topology (structure)|topology]].
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| In this case, we would expect the homomorphism ''f'' to preserve this additional structure; in the topological examples, we would want ''f'' to be a [[continuous map]].
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| The process may run into a snag with the quotient algebras, which may not be well-behaved.
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| 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).
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| ==Kernels in category theory==
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| The notion of ''kernel'' in [[category theory]] is a generalisation of the kernels of abelian algebras; see [[Kernel (category theory)]].
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| The categorical generalisation of the kernel as a congruence relation is the ''[[kernel pair]]''.
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| (There is also the notion of [[difference kernel]], or binary [[equalizer (mathematics)|equaliser]].)
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Kernel (Algebra)}}
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| [[Category:Algebra]]
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| [[Category:Isomorphism theorems]]
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| [[Category:Linear algebra]]
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