Kernel (algebra) - Citizendia

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector

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. ↔ 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. In Abstract algebra, the Fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the

In this article, we first survey kernels for some important types of algebraic structures; then we give general definitions from universal algebra for generic algebraic structures. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models"

1 Survey of examples
2 Universal algebra
- 2.1 General case
- 2.2 Mal'cev algebras

Survey of examples

Linear operators

Let V and W be vector spaces and let T be a linear transformation from V to W. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that If 0_W is the zero vector of W, then the kernel of T is the preimage of the singleton set {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. In Linear algebra, the null vector or zero vector is the vector (0 0 &hellip 0 in Euclidean space, all of whose components are zero In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Mathematics, a singleton is a set with exactly one element The kernel is usually denoted as "ker T ", or some variation thereof:

$\mathop{\mathrm{ker}}\, T := \{\mathbf{v} \in V : T\mathbf{v} = \mathbf{0}_{W}\}\mbox{.} \!$

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. The concept of a linear subspace (or vector subspace) is important in Linear algebra and related fields of Mathematics. Thus, it makes sense to speak of the quotient space V /(ker T ). In Linear algebra, the quotient of a Vector space V by a subspace N is a vector space obtained by "collapsing" N 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). In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage As a consequence, the dimension of V equals the dimension of the kernel plus the dimension of the image. In Mathematics, the dimension of a Vector space V is the cardinality (i

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 Mathematics, the dimension of a Vector space V is the cardinality (i Basis vector redirects here For basis vector in the context of crystals see Crystal structure. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, a system of linear equations (or linear system) is a collection of Linear equations involving the same set of Variables For example 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 the rank-nullity theorem. The column rank of a matrix A is the maximal number of Linearly independent columns of A.

Solving homogeneous differential equations often amounts to computing the kernel of certain differential operators. A homogeneous differential equation has several distinct meanings In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator For instance, in order to find all twice-differentiable functions f from the real line to itself such that

xf ''(x) + 3f '(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

(Tf )(x) = xf ''(x) + 3f '(x) - f (x)

for f in V and x an arbitrary real number. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a In Mathematics, the real numbers may be described informally in several different ways 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. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real This includes kernels for homomorphisms between abelian groups as a special case. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the This example captures the essence of kernels in general abelian categories; see Kernel (category theory). In Mathematics, an abelian category is a category in which Morphisms and objects can be added and in which kernels and Cokernels exist In Category theory and its applications to other branches of Mathematics, kernels are a generalization of the kernels of Group homomorphisms and the kernels

Group homomorphisms

Let G and H be groups and let f be a group homomorphism from G to H. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function 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 . In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that 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 a subgroup of G but in fact a normal subgroup. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. Thus, it makes sense to speak of the quotient group G /(ker f ). In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G 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 Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage

In the special case of abelian groups, this works in exactly the same way as in the previous section. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the

Ring homomorphisms

Let R and S be rings (assumed unital) and let f be a ring homomorphism from R to S. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication If 0_S is the zero 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. In Mathematics, a zero element is one of several generalizations of the number zero to other Algebraic structures These alternate meanings may or may not 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 a subring of R since it may not contain the multiplicative identity, it is nevertheless a two-sided ideal of R. In Mathematics, a subring is a Subset of a ring, which contains the Multiplicative identity and is itself a ring under the same Binary operations In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. Thus, it makes sense to speak of the quotient ring R/(ker f). In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the 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). In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage

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). In Abstract algebra a bimodule is an Abelian group that is both a left and a right module, such that the left and right multiplications are compatible In Mathematics, the codomain, or target, of a function f: X → Y is the set

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. In Mathematics, a Malcev algebra (or Maltsev algebra or Moufang–Lie algebra) over a field is a (nonassociative algebra that is antisymmetric so that

Monoid homomorphisms

Let M and N be monoids and let f be a monoid homomorphism from M to N. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation 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. In Mathematics, one can often define a direct product of objectsalready known giving a new one In Mathematics, an ordered pair is a collection of two distinguishable objects one of which is identified as the first coordinate (or the first entry The kernel is usually denoted "ker f" (or a variation). In symbols:

$\mathop{\mathrm{ker}} f := \{(m,m') \in M \times M : f(m) = f(m')\}\mbox{.} \!$

Since f is a function, the elements of the form (m,m) must belong to the kernel. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function 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. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" See Congruence (geometry for the term as used in elementary geometry Thus, it makes sense to speak of the quotient monoid M/(ker f). In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation 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). In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation

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. Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models"

General case

Let A and B be algebraic structures of a given type and let f be a homomorphism of that type from A to B. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector 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. In Mathematics, one can often define a direct product of objectsalready known giving a new one In Mathematics, an ordered pair is a collection of two distinguishable objects one of which is identified as the first coordinate (or the first entry The kernel is usually denoted "ker f" (or a variation). In symbols:

$\mathop{\mathrm{ker}} f := \{(a,a') \in A \times A : f(a) = f(a')\}\mbox{.} \!$

Since f is a function, the elements of the form (a,a) must belong to the kernel. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function The homomorphism f is injective if and only if its kernel is only the diagonal set {(a,a) : a in A}.

It turns out that ker f is an equivalence relation on A, and in fact a congruence relation. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" See Congruence (geometry for the term as used in elementary geometry Thus, it makes sense to speak of the quotient algebra A/(ker f). In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the 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). In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Algebra (mathematics, the word "algebra" usually means a Vector space or module equipped with an additional bilinear operation

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. In Mathematics, the kernel of a function f may be taken to be either the Equivalence relation on the function's domain

Mal'cev algebras

In the case of Mal'cev algebras, this construction can be simplified. In Mathematics, a Malcev algebra (or Maltsev algebra or Moufang–Lie algebra) over a field is a (nonassociative algebra that is antisymmetric so that Every Mal'cev algebra has a special neutral element (the zero vector in the case of vector spaces, the identity element in the case of groups, and the zero element in the case of rings or modules). In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that In Linear algebra, the null vector or zero vector is the vector (0 0 &hellip 0 in Euclidean space, all of whose components are zero In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a zero element is one of several generalizations of the number zero to other Algebraic structures These alternate meanings may or may not In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars 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. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X

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 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. In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Mathematics, a singleton is a set with exactly one element 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-sided ring ideal in the case of rings, and submodule in the case of modules). The concept of a linear subspace (or vector subspace) is important in Linear algebra and related fields of Mathematics. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars It turns out that although ker f may not be a subalgebra of A, it is nevertheless an ideal. In Algebra (mathematics, the word "algebra" usually means a Vector space or module equipped with an additional bilinear operation In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. Then it makes sense to speak of the quotient algebra G/(ker f). In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the 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 of quotient in the Mal'cev algebra (which is division on either side for groups and subtraction for vector spaces, modules, and rings). In Mathematics, a quotient is the result of a division. For example when dividing 6 by 3 the quotient is 2 while 6 is called the dividend, and 3 the In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. Subtraction is one of the four basic Arithmetic operations it is the inverse of Addition, meaning that if we start with any number and add any number and then subtract 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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