# Kernel (algebra)

In the various branches of

mathematics that fall under the heading ofabstract algebra , the**kernel**of ahomomorphism measures the degree to which the homomorphism fails to beinjective . An important special case is the kernel of a matrix, also called the "null space".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 isinjective . Thefundamental theorem on homomorphisms (orfirst isomorphism theorem ) is a theorem, again taking various forms, that applies to thequotient algebra defined by the kernel.In this article, we first survey kernels for some important types of

algebraic structure s; then we give general definitions fromuniversal algebra for generic algebraic structures.**Survey of examples****Linear operators**Let "V" and "W" be

vector space s and let "T" be alinear transformation from "V" to "W". If**0**_{"W"}is thezero vector of "W", then the kernel of "T" is thepreimage of thesingleton set {**0**_{"W"}}; that is, thesubset of "V" consisting of all those elements of "V" that are mapped by "T" to the element**0**_{"W"}. The kernel is usually denoted as "ker "T" ", or some variation thereof::$mathop\{mathrm\{ker,\; T\; :=\; \{mathbf\{v\}\; in\; V\; :\; Tmathbf\{v\}\; =\; mathbf\{0\}\_\{W\}\}mbox\{.\}\; !$

Since a linear transformation preserves zero vectors, the zero vector

**0**_{"V"}of "V" must belong to the kernel. The transformation "T" is injective if and only if its kernel is only the singleton set {**0**_{"V"}}.It turns out that ker "T" is always a

linear subspace of "V". Thus, it makes sense to speak of the quotient space "V" /(ker "T" ). The first isomorphism theorem for vector spaces states that this quotient space is naturally isomorphic to the image of "T" (which is a subspace of "W"). As a consequence, the dimension of "V" equals the dimension of the kernel plus the dimension of the image.If "V" and "W" are finite-dimensional and bases have been chosen, then "T" can be described by a matrix "M", and the kernel can be computed by solving the homogeneous

system of linear equations "M"**v**=**0**. In this representation, the kernel corresponds to the 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 of "M", as a consequence of therank-nullity theorem .Solving

homogeneous differential equation s often amounts to computing the kernel of certaindifferential operator s.For instance, in order to find all twice-differentiable function s*f*from thereal line to itself such that:*x**f*" (*x*) + 3*f*'(*x*) =*f*(*x*),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: (*T**f*)(*x*) =*x**f*" (*x*) + 3*f*'(*x*) -*f*(*x*)for*f*in*V*and*x*an arbitraryreal number .Then all solutions to the differential equation are in ker*T*.One can define kernels for

homomorphism s between modules over a ring in an analogous manner.This includes kernels for homomorphisms betweenabelian group s as a special case.This example captures the essence of kernels in generalabelian categories ; seeKernel (category theory) .**Group homomorphisms**Let

*G*and*H*be groups and let*f*be agroup homomorphism from*G*to*H*.If*e*_{H}is theidentity element of*H*, then the "kernel" of*f*is the preimage of the singleton set {*e*_{H}}; that is, the subset of*G*consisting of all those elements of*G*that are mapped by*f*to the element*e*_{H}.The kernel is usually denoted "ker*f*" (or a variation).In symbols:: $mathop\{mathrm\{ker\; f\; :=\; \{g\; in\; G\; :\; f(g)\; =\; e\_\{H\}\}mbox\{.\}\; !$Since a group homomorphism preserves identity elements, the identity element

*e*_{G}of*G*must belong to the kernel.The homomorphism*f*is injective if and only if its kernel is only the singleton set {*e*_{G}}.It turns out that ker

*f*is not only asubgroup of*G*but in fact anormal subgroup .Thus, it makes sense to speak of thequotient group *G*/(ker*f*).The first isomorphism theorem for groups states that this quotient group is naturally isomorphic to the image of*f*(which is a subgroup of*H*).In the special case of

abelian group s, this works in exactly the same way as in the previous section.**Ring homomorphisms**Let

*R*and*S*be rings (assumedunital ) and let*f*be aring homomorphism from*R*to*S*.If*0*_{S}is thezero element of*S*, then the "kernel" of*f*is the preimage of the singleton set {*0*_{S}}; that is, the subset of*R*consisting of all those elements of*R*that are mapped by*f*to the element*0*_{S}.The kernel is usually denoted "ker*f*" (or a variation).In symbols:: $mathop\{mathrm\{ker\; f\; :=\; \{r\; in\; R\; :\; f(r)\; =\; 0\_\{S\}\}mbox\{.\}\; !$Since a ring homomorphism preserves zero elements, the zero element

*0*_{R}of*R*must belong to the kernel.The homomorphism*f*is injective if and only if its kernel is only the singleton set {*0*_{R}}.It turns out that, although ker

*f*is generally not asubring of*R*since it may not contain the multiplicative identity, it is nevertheless a two-sided ideal of*R*.Thus, it makes sense to speak of thequotient ring *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*).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*:

**R*itself;

* any two-sided ideal of*R*(such as ker*f*);

* any quotient ring of*R*(such as*R*/(ker*f*)); and

* thecodomain of any ring homomorphism whose domain is*R*(such as*S*, the codomain of*f*).However, the isomorphism theorem gives a stronger result, because ring isomorphisms preserve multiplication while module isomorphisms (even between rings) in general do not.This example captures the essence of kernels in general

Mal'cev algebra s.**Monoid homomorphisms**Let

*M*and*N*be monoids and let*f*be amonoid homomorphism from*M*to*N*.Then the "kernel" of*f*is the subset of thedirect product *M*×*M*consisting of all thoseordered pair s of elements of*M*whose components are both mapped by*f*to the same element in*N*.The kernel is usually denoted "ker*f*" (or a variation).In symbols:: $mathop\{mathrm\{ker\; f\; :=\; \{(m,m\text{'})\; in\; M\; imes\; M\; :\; f(m)\; =\; f(m\text{'})\}mbox\{.\}\; !$Since

*f*is a function, the elements of the form (*m*,*m*) must belong to the kernel.The homomorphism*f*is injective if and only if its kernel is only thediagonal set {(m,m) :*m*in*M*}.It turns out that ker

*f*is anequivalence relation on*M*, and in fact acongruence relation .Thus, it makes sense to speak of thequotient monoid *M*/(ker*f*).The first isomorphism theorem for monoids states that this quotient monoid is naturally isomorphic to the image of*f*(which is asubmonoid of*N*).This is very different in flavour from the above examples.In particular, the preimage of the identity element of

*N*is "not" enough to determine the kernel of*f*.This is because monoids are not Mal'cev algebras.**Universal algebra**All the above cases may be unified and generalized in

universal algebra .**General case**Let

*A*and*B*bealgebraic structure s of a given type and let*f*be ahomomorphism of that type from*A*to*B*.Then the "kernel" of*f*is the subset of thedirect product *A*×*A*consisting of all thoseordered pair s of elements of*A*whose components are both mapped by*f*to the same element in*B*.The kernel is usually denoted "ker*f*" (or a variation).In symbols:: $mathop\{mathrm\{ker\; f\; :=\; \{(a,a\text{'})\; in\; A\; imes\; A\; :\; f(a)\; =\; f(a\text{'})\}mbox\{.\}\; !$Since

*f*is a function, the elements of the form (*a*,*a*) must belong to the kernel.The homomorphism*f*is injective if and only if its kernel is only thediagonal set {(a,a) :*a*in*A*}.It turns out that ker

*f*is anequivalence relation on*A*, and in fact acongruence relation .Thus, it makes sense to speak of thequotient algebra *A*/(ker*f*).The first isomorphism theorem in general universal algebra states that this quotient algebra is naturally isomorphic to the image of*f*(which is asubalgebra of*B*).Note that the definition of kernel here (as in the monoid example) doesn't depend on the algebraic structure; it is a purely set-theoretic concept.For more on this general concept, outside of abstract algebra, see

kernel of a function .**Mal'cev algebras**In the case of Mal'cev algebras, this construction can be simplified. Every Mal'cev algebra has a special

neutral element (thezero vector in the case ofvector space s, theidentity element in the case of groups, and thezero element in the case of rings or modules). The characteristic feature of a Mal'cev algebra is that we can recover the entire equivalence relation ker*f*from theequivalence class of the neutral element.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*_{B}is the neutral element of*B*, then the "kernel" of*f*is thepreimage of thesingleton set {*e*_{B}}; that is, thesubset of*A*consisting of all those elements of*A*that are mapped by*f*to the element*e*_{B}.The kernel is usually denoted "ker*f*" (or a variation). In symbols:: $mathop\{mathrm\{ker\; f\; :=\; \{a\; in\; A\; :\; f(a)\; =\; e\_\{B\}\}mbox\{.\}\; !$Since a Mal'cev algebra homomorphism preserves neutral elements, the identity element

*e*_{A}of*A*must belong to the kernel. The homomorphism*f*is injective if and only if its kernel is only the singleton set {*e*_{A}}.The notion of ideal generalises to any Mal'cev algebra (as

linear subspace in the case of vector spaces,normal subgroup in the case of groups, two-sidedring ideal in the case of rings, andsubmodule in the case of modules). It turns out that although ker*f*may not be asubalgebra of*A*, it is nevertheless an ideal.Then it makes sense to speak of thequotient algebra *G*/(ker*f*).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*).The connection between this and the congruence relation is for more general types of algebras is as follows.First, the kernel-as-an-ideal is the equivalence class of the neutral element

*e*_{A}under the kernel-as-a-congruence. For the converse direction, we need the notion ofquotient in the Mal'cev algebra (which is division on either side for groups andsubtraction for vector spaces, modules, and rings).Using this, elements*a*and*a*' of*A*are equivalent under the kernel-as-a-congruence if and only if their quotient*a*/*a*' is an element of the kernel-as-an-ideal.

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