Kernel (algebra)

Group homomorphisms A group is a set G with a binary operation \cdot satisfying the following three properties for all a,b,c \in G: • Associative: (a \cdot b) \cdot c = a \cdot (b \cdot c) • Identity: There is an e \in G such that e \cdot a = a \cdot e = a • Inverses: There is an a' \in G for each a \in G such that a \cdot a' = a' \cdot a = e A group is also called abelian if it also satisfies a \cdot b = b \cdot a. (For simplicity, the operation symbol \cdot is omitted.) Letting e_H be 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). \ker{f} is a subgroup of G and further it is a normal subgroup. Thus, there is a corresponding quotient group G/\ker{f}. This is isomorphic to f(G), the image of G under f (which is a subgroup of H also), by the first isomorphism theorem for groups. • R with + is an abelian group with identity 0. • Multiplication \cdot is associative. • Distributive: a \cdot (b + c) = a \cdot b + a \cdot c and (a + b) \cdot c = a \cdot c + b \cdot c for all a,b,c \in R • Multiplication \cdot has an identity element 1. A ring is commutative if the multiplication is commutative, and such a ring is a field when every 0 \neq a \in R has a multiplicative inverse, that is, some b \in R where ab=1. • f(a+b)=f(a)+f(b) • f(ab)=f(a)f(b) The kernel of f is the kernel as additive groups. It is the preimage of the zero ideal \{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: : \operatorname{ker} f = \{r \in R : f(r) = 0_{S}\} . 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. • a(b \alpha) = (ab) \alpha • (a+b)\alpha = a\alpha + b\alpha • a(\alpha + \beta) = a\alpha + a\beta • 1\alpha = \alpha Let V and W be vector spaces over the field F. A linear map (or linear transformation) from V to W is a function T: V \to W satisfying for all \alpha, \beta \in V and a \in F: • T(\alpha + \beta) = T(\alpha)+T(\beta) • T(a \alpha) = a T(\alpha) If 0_W is the zero vector of W, then the kernel of T (or null space 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. Let M and N be R-modules. A module homomorphism from M to N is also a function \varphi: M \to N satisfying the same analogous properties of a linear map. The kernel of \varphi is defined to be: : \ker \varphi = \{m \in M \ | \ \varphi (m) = 0\} Every kernel is a submodule of the domain module, which means they always contain 0, the additive identity of the module. Kernels of abelian groups can be considered a particular kind of module kernel when the underlying ring is the integers. == Examples ==

Group homomorphisms 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 \in H. The quotient group G / \ker{f} has two elements: \{0,2,4\} and \{1,3,5\}, and is isomorphic to H. Given a isomorphism \varphi: G \to H, one has \ker \varphi = 1. Then the function f: \mathbb{R} \to S^1 sending x \mapsto e^{2\pi ix}=\cos(2\pi x)+i\sin(2\pi x) is a homomorphism with the integers being the kernel. The first isomorphism theorem then implies that \mathbb{R}/\mathbb{Z} \cong S^1. The symmetric group on n elements, S_n, has a surjective homomorphism \epsilon: S_n \to \mathbb{Z}_2 that takes each permutation to the parity of the number of transpositions whose product is that permutation. The alternating group A_n = \ker \epsilon is the kernel of this homomorphism, consisting of the even permutations. The alternating group is a non-abelian simple group for n \geq 5. The determinant of n \times n invertible matrices of the real numbers \mathbb{R}, whose set is denoted GL(n, \mathbb{R}) and called the general linear group of n \times n matrices of \mathbb{R}, is a homomorphism onto the multiplication group \mathbb{R}^\times (consisting of all non-zero real numbers), and the kernel of the determinant is called the special linear group SL(n,\mathbb{R}) of n \times n matrices of \mathbb{R}. These are the matrices whose determinant is precisely 1. Given a group G and an element, the mapping x \mapsto gxg^{-1} is an automorphism - an isomorphism whose domain and image are the same group. This gives a homomorphism from G to its automorphism group \text{Aut}(G), mapping each g to its respective inner automorphism as described, and the kernel of this homomorphism is the center Z(G) of G, consisting of g \in G where for every x \in G, we have gxg^{-1}=x, or equivalently gx=xg. More generally, for every normal subgroup H of G (i.e. groups closed under conjugation), this conjugation map is also an automorphism on H, giving another homomorphism G to \text{Aut}(H), with the kernel being the centralizer C_G(H) of H in G, being the set of g \in G where for every h \in H, we have ghg^{-1}=h. Ring homomorphisms Consider the mapping \varphi : \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} where the later ring is the integers modulo 2 and the map sends each number to its 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. Let \varphi: \mathbb{Q}[x] \to \mathbb{Q} be defined as \varphi(p(x))=p(0). 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. Linear maps Let \varphi: \mathbb{C}^3 \to \mathbb{C} be defined as \varphi(x,y,z) = x+2y+3z, then the kernel of \varphi (that is, the null space) will be the set of points (x,y,z) \in \mathbb{C}^3 such that x+2y+3z=0, and this set is a subspace of \mathbb{C}^3 (the same is true for every kernel of a linear map). If D represents the derivative operator on real polynomials, then the kernel of D will consist of the polynomials with deterivative equal to 0, that is the constant functions. Consider the mapping (Tp)(x)=x^2p(x), where p is a polynomial with real coefficients. Then T is a linear map whose kernel is precisely 0, since 0 is the only polynomial to satisfy x^2p(x) = 0 for all x \in \mathbb{R}. == Quotient algebras ==

The kernel of a homomorphism can be used to define a quotient algebra. Let G and H be groups, \varphi: G \to H be a group homomorphism, and denote K = \ker \varphi . Put G/K to be the set of fibers of the homomorphism \varphi, where a fiber is the set of points of the domain mapping to a single point in the range. Let X_a \in G/K denotes the fiber of the element a \in H , then a group operation on the set of fibers can be endowed by X_a X_b = X_{ab}, and G/K is called the quotient group (or factor group), to be read as "G modulo K" or "G mod K". The group operation can then be defined as uK \circ vK = (uk)K, which is well-defined regardless of the choice of representatives of the fibers. According to the first isomorphism theorem, there is an isomorphism \mu: G/K \to \varphi(G), where the later group is the image of the homomorphism \varphi, and the isomorphism is defined as \mu(uK)=\varphi(u), and such map is also well-defined. For rings, modules, and vector spaces, one can define the respective quotient algebras via the underlying additive group structure, with cosets represented as x+K. Ring multiplication can be defined on the quotient algebra as (x+K)(y+K)=xy+K, and is well-defined. == Kernel structures ==

The structure of kernels allows for the building of quotient algebras from structures satisfying the properties of kernels. Any subgroup N of a group G can construct a quotient G/N by the set of all cosets of N in G. 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 uN \cdot vN = (uv)N, however this operation is well defined if and only if the subgroup N is closed under conjugation under G, that is, if g \in G and n \in N, then gng^{-1} \in N. Furthermore, the operation being well defined is sufficient for the quotient to be a group. Subgroups satisfying this property are called normal subgroups. Every kernel of a group is a normal subgroup, and for a given normal subgroup N of a group G, the natural projection \pi: G \to G/N defined as \pi(g) = gN is a homomorphism with \ker \pi = N, so the normal subgroups are precisely the subgroups which are kernels. The closure under conjugation, however, gives a criterion for when a subgroup is a kernel for some homomorphism. For a ring R, treating it as a group, one can take a quotient group via an arbitrary subgroup I of the ring, which will be normal due to the ring's additive group being abelian. To define multiplication on R/I, the multiplication of cosets, defined as (r+I)(s+I) = rs + I needs to be well-defined. Taking representatives r+\alpha and s+\beta of r + I and s + I respectively, for r,s \in R and \alpha, \beta \in I, yields: : (r + \alpha)(s + \beta) + I = rs + I Setting r = s = 0 implies that I is closed under multiplication, while setting \alpha = s = 0 shows that r\beta \in I, that is, I is closed under arbitrary multiplication by elements on the left. Similarly, taking r = \beta = 0 implies that I is also closed under multiplication by arbitrary elements on the right. Any subgroup of R that is closed under multiplication by any element of the ring is called an ideal. Analogously to normal subgroups, the ideals of a ring are precisely the kernels of homomorphisms. == Exact sequence ==

Kernels are used to define exact sequences of homomorphisms for groups and modules. Given modules A, B, and C, a pair of homomorphisms \psi: A \to B, \varphi: B \to C, written as A \xrightarrow{\psi} B \xrightarrow{\varphi} C is said to be exact (at B) if \text{image } \psi = \ker \varphi. An exact sequence is then a sequence of modules and homomorphisms \cdots \to X_{n-1} \to X_n \to X_{n+1} \to \cdots where each adjacent pair of modules and homomorphisms is exact. It is unnecessary to label the homomorphisms in an exact sequence which start or end at the zero module as there is only one unique map; the map 0 \mapsto 0 when the zero module is the domain, and the map b \mapsto 0 when the zero module is the range. Exact sequences can be used to describe when a homomorphism is injective, surjective, or an isomorphism. In particular, the sequences 0 \to A \xrightarrow{f} B, B \xrightarrow{g} C \to 0, and 0 \to A \xrightarrow{h} B \to 0 are exact if and only if the labeled homomorphism are injective, surjective, and an isomorphism respectively. A particular kind of exact sequence is a short exact sequence, which is of the form 0 \to A \xrightarrow{\psi} B \xrightarrow{\varphi} C \to 0. These sequences are related to the extension problem: given modules A and C, determine the modules B where A is a submodule of B, and their resulting quotient is isomorphic to C. Such a module is called an extension of C by A (or alternatively, an extension of A by C). The extension problem, when written as exact sequences, can be stated as finding all short exact sequences 0 \to A \xrightarrow{\psi} B \xrightarrow{\varphi} C \to 0 with A and C fixed. Such an extension implies that A \cong \psi(A) and \psi(A) is the kernel of . == Universal algebra ==

Kernels can be generalized in universal algebra for homomorphisms between any two algebraic structures. An operation on a set A is a function of the form Q:A^n \to A, where n is the arity (or rank) of the operation. An n-ary operation takes an ordered list of n elements from A and maps them to a single element in A. An algebraic structure is a tuple \langle A, F \rangle where A is the underlying set of the algebra, and F is an indexed set of operations Q \in F on A, with their interpretation denoted Q^A. The set indexing F is the language, which also maps each operation symbol to their fixed arity (called the rank function). Two algebraic structures are similar when they share the same language, including their rank function. Let A and B be algebraic structures of a similar type F. A homomorphism is a function f: A \to B that respects the interpretation of each Q \in F, that is, taking Q to be an n-ary operation, and a_i \in A for 1 \leq i \leq n: : f(Q^A(a_1, \ldots a_n)) = Q^B(f(a_1), \ldots f(a_n)) The kernel of f, denoted \ker{f}, is the subset of the direct product A \times A consisting of all ordered pairs of elements of A whose components are both mapped by f to the same element in B. In symbols: : Q^{A/\ker{f}}(a_1/\ker{f}, \ldots a_n/\ker{f}) = Q^A(a_1, \ldots a_n)/\ker{f} The first isomorphism theorem in universal algebra states that this quotient algebra is naturally isomorphic to the image of f (which is a subalgebra of B). == Category theory ==

Kernels of morphisms Kernels can be generalized in categories that have zero objects. A category must satisfy having: • Objects A \in \bold{C} • Morphisms f: A \to B • Composition; if f: A \to B and g: B \to C, then denote their composition as g \circ f: A \to C • Associativity: if f: A \to B, g: B \to C, and h: C \to D, then h \circ (g \circ f) = (h \circ g) \circ f • An identity morphism id_A: A \to A where composition with it results in the same morphism; for f: A \to B, f = f \circ id_A = id_B \circ f A morphism f: A \to B is an isomorphism when there exists a morphism g: B \to A such that g \circ f and f \circ g are the identity morphisms. If the zero object of a category is labeled 0, then the composition of the morphisms 0: A \to 0 \to B is the 0-morphism from A to B. The kernel is denoted as \ker f \to B. The kernel is the limit of the diagram B \xrightarrow{f} C \xleftarrow{} 0. By reversing the direction of the morphisms and compositions given in the definition of a kernel, this defines the notion of a cokernel, denoted as \text{coker} f. The image (category theory) of a morphism is defined as \text{im} f = \ker (\text{coker} f) when the respective kernel/cokernel exist. For abelian groups, the equalizer of two homomorphisms is the same as the equalizer between the difference of these two homomorphisms and the zero homomorphism, so the only equalizers that are needed to be considered in the category of abelian groups are the ones between any homomorphism h: A \to B and the zero homomorphism 0: A \to B. The object of such an equalizer is (up to isomorphism) \ker h, the kernel of the homomorphism h, and the associated morphism is the inclusion map. Kernel pairs The kernel pair of a morphism f: X \to Y is defined as the pullback on this morphism paired with itself. It can be visualized with the commutative diagram: Kernels of functors Functors between categories can also have a kernel. A (covariant) functor from a category \bold{C} to \bold{D}, denoted F: \bold{C} \to \bold{D}, maps objects and morphisms from \bold{C} to \bold{D} such that the following hold: • If f: A \to B, then F(f): F(A) \to F(B) • F(g \circ f) = F(g) \circ F(f) • F(id_A) = id_{F(A)} A congruence on a category \bold{C} is an equivalence relation \sim on morphisms where f \sim g implies they share the same domain and codomain, and furthermore bfa \sim bga for any applicable morphisms a and b. A congruence gives rise to an associated congruence category \bold{C}^\sim with the same objects as \bold{C} but with morphisms consisting of \langle f, g \rangle where f \sim g, composition being defined componentwise, and the identity morphism being \widetilde{id_A} = \langle id_A, id_A \rangle. Then a quotient category \bold{C}/\sim can be formed, where the objects are again the same as \bold{C}, the morphisms are the equivalence classes [f] under the congruence, the identity morphism being its associated equivalence class [id_A], and composition defined as [g] \circ [f] = [g \circ f]. There are two projection functors from the congruence category to the original category, labeled as p_1, p_2, and there is a quotient functor \pi from the category to its quotient category acting as the coequalizer of the two projection functors. A functor F: \bold{C} \to \bold{D} gives a congruence \sim_F where f \sim_F g if and only if they share the same domain and codomain, and furthermore F(f) = F(g). The kernel of F is then denoted as the associated congruence category \ker F = \bold{C}^{\sim_F}. == See also ==