In algebra, the **kernel** of a homomorphism (function that preserves the structure) is generally the inverse image of 0 (except for groups whose operation is denoted multiplicatively, where the kernel is the inverse image of 1). An important special case is the kernel of a linear map. The kernel of a matrix, also called the *null space*, is the kernel of the linear map defined by the matrix.

- Survey of examples
- Linear maps
- Group homomorphisms
- Ring homomorphisms
- Monoid homomorphisms
- Universal algebra
- General case
- Malcev algebras
- Algebras with nonalgebraic structure
- Kernels in category theory
- See also
- Notes
- References

The kernel of a homomorphism is reduced to 0 (or 1) if and only if the homomorphism is 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.^{ [1] }

For some types of structure, such as abelian groups and vector spaces, 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 ideals for rings.

Kernels allow defining quotient objects (also called quotient algebras in universal algebra, and cokernels in category theory). For many types of algebraic structure, the fundamental theorem on homomorphisms (or first isomorphism theorem) states that image of a homomorphism is 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 an 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.

Let *V* and *W* be vector spaces over a field (or more generally, modules over a ring) and let *T* be a linear map from *V* to *W*. If **0**_{W} is the zero vector of *W*, then the kernel of *T* is the preimage of the zero subspace {**0**_{W}}; that is, the subset 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:

Since a linear map 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 reduced to the zero subspace.

The kernel 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 case, the kernel of *T* may be identified to the 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 of *M*, as a consequence of the rank–nullity theorem.

Solving homogeneous differential equations often amounts to computing the kernel of certain differential operators. For instance, in order to find all twice-differentiable functions *f* from the real line to itself such that

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

for *f* in *V* and *x* an arbitrary real number. Then all solutions to the differential equation are in ker *T*.

One can define kernels for homomorphisms between modules over a ring in an analogous manner. This includes kernels for homomorphisms between abelian groups as a special case. This example captures the essence of kernels in general abelian categories; see Kernel (category theory).

Let *G* and *H* be groups and let *f* be a group homomorphism from *G* to *H*. If *e*_{H} is the identity 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:

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}}. This is true because if the homomorphism *f* is not injective, then there exists with such that . This means that , which is equivalent to stating that since group homomorphisms carry inverses into inverses and since . In other words, . Conversely, if there exists an element , then , thus *f* is not injective.

It turns out that ker *f* is not only a subgroup of *G* but in fact a normal subgroup. Thus, it makes sense to speak of the quotient 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 groups, this works in exactly the same way as in the previous section.

Let *G* be the cyclic group on 6 elements {0,1,2,3,4,5} with modular addition, *H* be the cyclic on 2 elements {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 ker *f* = {0, 2, 4}, since all these elements are mapped to 0_{H}. The quotient group *G*/(ker *f*) has two elements: {0,2,4} and {1,3,5}. It is indeed isomorphic to *H*.

Algebraic structure → Ring theoryRing theory |
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Let *R* and *S* be rings (assumed unital) and let *f* be a ring homomorphism from *R* to *S*. If 0_{S} 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_{S}}, which 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:

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}}. This is always the case if *R* is a 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 subrng, and, more precisely, a two-sided ideal of *R*. Thus, it makes sense to speak of the quotient 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*). (note that rings need not be unital for the kernel definition).

To some extent, this can be thought of as a special case of the situation for modules, since these are all bimodules 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 - the codomain 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 algebras.

Let *M* and *N* be monoids and let *f* be a monoid homomorphism from *M* to *N*. Then the *kernel* of *f* is the subset of the direct product *M*×*M* consisting of all those ordered pairs 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:

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 the diagonal set {(m,m) : *m* in *M*}.

It turns out that ker *f* is an equivalence relation on *M*, and in fact a congruence relation. Thus, it makes sense to speak of the quotient 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 a submonoid of *N*),(for the congruence relation).

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*.

All the above cases may be unified and generalized in universal algebra.

Let *A* and *B* be algebraic structures 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 *A*×*A* consisting of all those ordered pairs 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:

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 exactly the diagonal set {(*a*,*a*) : *a*∈*A*}.

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 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 a subalgebra 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.

This section may be confusing or unclear to readers. In particular, this section cannot be understood, as referring to a structure which is different from Malcev algebra and is not defined nor linked.(December 2016) (Learn how and when to remove this template message) |

In the case of Malcev algebras, this construction can be simplified. Every Malcev algebra has a special neutral element (the zero vector in the case of vector spaces, the identity element in the case of commutative groups, and the zero element in the case of rings or modules). The characteristic feature of a Malcev algebra is that we can recover the entire equivalence relation ker *f* from the equivalence class of the neutral element.

To be specific, let *A* and *B* be Malcev 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 the preimage of the singleton set {*e*_{B}}; that is, the subset 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:

Since a Malcev 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 Malcev 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 modules). It turns out that ker *f* is not a subalgebra of *A*, but it is an ideal. Then it makes sense to speak of the quotient algebra *G*/(ker *f*). The first isomorphism theorem for Malcev 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 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 of quotient in the Mal'cev algebra (which is division on either side for groups and subtraction for vector spaces, modules, and rings). 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.

Sometimes algebras are equipped with a nonalgebraic structure in addition to their algebraic operations. For example, one may consider topological groups or topological vector spaces, with are equipped with a topology. 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. 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 (as is usually done); then the kernel (however it is constructed) will be a closed set and the quotient space will work fine (and also be Hausdorff).

The notion of *kernel* in category theory is a generalisation of the kernels of abelian algebras; see Kernel (category theory). The categorical generalisation of the kernel as a congruence relation is the * kernel pair *. (There is also the notion of difference kernel, or binary equaliser.)

- ↑ See Dummit & Foote 2004 and Lang 2002.

A **quotient group** or **factor group** is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure. For example, the cyclic group of addition modulo *n* can be obtained from the group of integers under addition by identifying elements that differ by a multiple of *n* and defining a group structure that operates on each such class as a single entity. It is part of the mathematical field known as group theory.

In abstract algebra, the **fundamental theorem on homomorphisms**, also known as the **fundamental homomorphism theorem**, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.

In mathematics, given two groups, and, a **group homomorphism** from to is a function *h* : *G* → *H* such that for all *u* and *v* in *G* it holds that

In algebra, a **homomorphism** is a structure-preserving map between two algebraic structures of the same type. The word *homomorphism* comes from the Ancient Greek language: ὁμός meaning "same" and μορφή meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German *ähnlich* meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

In ring theory, a branch of abstract algebra, a **ring homomorphism** is a structure-preserving function between two rings. More explicitly, if *R* and *S* are rings, then a ring homomorphism is a function *f* : *R* → *S* such that *f* is

In mathematics, specifically abstract algebra, the **isomorphism theorems** are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.

In abstract algebra, a **congruence relation** is an equivalence relation on an algebraic structure that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes for the relation.

In mathematics, **rings** are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a *ring* is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

An **exact sequence** is a sequence of morphisms between objects such that the image of one morphism equals the kernel of the next.

In ring theory, a branch of abstract algebra, a **quotient ring**, also known as **factor ring**, **difference ring** or **residue class ring**, is a construction quite similar to the quotient groups of group theory and the quotient spaces of linear algebra. It is a specific example of a quotient, as viewed from the general setting of universal algebra. One starts with a ring *R* and a two-sided ideal *I* in *R*, and constructs a new ring, the quotient ring *R* / *I*, whose elements are the cosets of *I* in *R* subject to special *+* and *⋅* operations.

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.

In mathematics, a **module** is one of the fundamental algebraic structures used in abstract algebra. A **module over a ring** is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring and a multiplication is defined between elements of the ring and elements of the module. A module taking its scalars from a ring *R* is called an *R*-module.

In mathematics, a **quotient algebra** is the result of partitioning the elements of an algebraic structure using a congruence relation. Quotient algebras are also called **factor algebras**. Here, the congruence relation must be an equivalence relation that is additionally *compatible* with all the operations of the algebra, in the formal sense described below. Its equivalence classes partition the elements of the given algebraic structure. The quotient algebra has these classes as its elements, and the compatibility conditions are used to give the classes an algebraic structure.

In set theory, the **kernel** of a function *f* may be taken to be either

In mathematics, the **Gelfand representation** in functional analysis has two related meanings:

In algebra, a **module homomorphism** is a function between modules that preserves the module structures. Explicitly, if *M* and *N* are left modules over a ring *R*, then a function is called an *R*-*module homomorphism* or an *R*-*linear map* if for any *x*, *y* in *M* and *r* in *R*,

In mathematics, the **symmetric algebra***S*(*V*) on a vector space *V* over a field *K* is a commutative algebra over K that contains V, and is, in some sense, minimal for this property. Here, "minimal" means that *S*(*V*) satisfies the following universal property: for every linear map f from V to a commutative algebra A, there is a unique algebra homomorphism *g* : *S*(*V*) → *A* such that *f* = *g* ∘ *i*, where i is the inclusion map of V in *S*(*V*).

In linear algebra, the **quotient** of a vector space *V* by a subspace *N* is a vector space obtained by "collapsing" *N* to zero. The space obtained is called a **quotient space** and is denoted *V*/*N*.

In mathematics, the **congruence lattice problem** asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed by Robert P. Dilworth, and for many years it was one of the most famous and long-standing open problems in lattice theory; it had a deep impact on the development of lattice theory itself. The conjecture that every distributive lattice is a congruence lattice is true for all distributive lattices with at most ℵ_{1} compact elements, but F. Wehrung provided a counterexample for distributive lattices with ℵ_{2} compact elements using a construction based on Kuratowski's free set theorem.

In mathematics, **Lie group–Lie algebra correspondence** allows one to study Lie groups, which are geometric objects, in terms of Lie algebras, which are linear objects. In this article, a Lie group refers to a real Lie group. For the complex and *p*-adic cases, see complex Lie group and *p*-adic Lie group.

- Dummit, David S.; Foote, Richard M. (2004).
*Abstract Algebra*(3rd ed.). Wiley. ISBN 0-471-43334-9.

- Lang, Serge (2002).
*Algebra*. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.

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