Kernel is the algebraic device that measures exactly what a structure-preserving map collapses. A homomorphism may identify different elements of its domain, and the kernel records the elements that become invisible after applying the map. For [groups](/page/Group), [rings](/page/Ring), [modules](/page/Module), and [linear maps](/page/Linear%20Map), this single idea explains why quotient objects arise naturally and why isomorphism theorems have their particular form. The guiding question is not merely whether a homomorphism loses information, but how that loss is organised. In a [group homomorphism](/page/Group%20Homomorphism), the elements sent to the identity form a [normal subgroup](/page/Normal%20Subgroup). In a ring homomorphism, the elements sent to zero form an ideal. In a module homomorphism, the elements sent to zero form a submodule. Kernels are therefore the bridge between maps and quotients: they turn the failure of injectivity into an internal algebraic object. ## Definition Before separating the group, ring, and module cases, it is useful to name the common pattern. A kernel belongs to a structure-preserving map and records the elements of the domain that are sent to the distinguished neutral element of the codomain. The exact neutral element depends on the structure: the identity for groups, additive zero for rings, and additive zero for modules. [definition: Kernel] Let $X$ and $Y$ be algebraic structures with a distinguished neutral element $e_Y \in Y$, and let $f: X \to Y$ be a structure-preserving map. The kernel of $f$ is the subset \begin{align*} \ker f = \{x \in X : f(x) = e_Y\}. \end{align*} [/definition] This general definition is a template rather than a replacement for the standard algebraic definitions. The value of the kernel comes from the extra closure properties forced by the kind of homomorphism under discussion. [example: Zero Map] Let $V$ be a [vector space](/page/Vector%20Space) and let $0:V\to V$ be the zero linear map. Every vector is sent to $0$, so \begin{align*} \ker 0=\{v\in V:0(v)=0\}=V. \end{align*} This example shows that a kernel can be the whole domain when the map forgets all information. [/example] ## Algebraic Forms ### Group Homomorphisms A homomorphism preserves a chosen algebraic structure, so the element that plays the role of zero information depends on the structure. For groups this element is the identity, because the identity is the element whose insertion has no observable effect on multiplication. The kernel isolates the part of the domain that becomes indistinguishable from that identity after the map is applied. [definition: Kernel of a Group Homomorphism] Let $G$ and $H$ be groups with identities $1_G$ and $1_H$, and let $\varphi: G \to H$ be a group homomorphism. The kernel of $\varphi$ is \begin{align*} \ker \varphi = \{g \in G : \varphi(g) = 1_H\}. \end{align*} [/definition] ### Ring Homomorphisms To handle rings, the definition needs to change from the identity element of a group to the additive zero of a ring. This version is needed because quotient rings are built by declaring certain elements to be zero while preserving addition and multiplication. Throughout this page, rings have multiplicative identities and ring homomorphisms preserve those identities. [definition: Kernel of a Ring Homomorphism] Let $R$ and $S$ be rings, and let $\varphi: R \to S$ be a ring homomorphism. The kernel of $\varphi$ is \begin{align*} \ker \varphi = \{r \in R : \varphi(r) = 0_S\}. \end{align*} [/definition] ### Module Homomorphisms To handle linear equations and module quotients, the same zero-fibre idea must respect scalar multiplication. This definition is needed because the homogeneous solution set of a module homomorphism should itself be stable under the module operations. [definition: Kernel of a Module Homomorphism] Let $R$ be a ring, let $M$ and $N$ be left $R$-modules, and let $f: M \to N$ be an $R$-module homomorphism. The kernel of $f$ is \begin{align*} \ker f = \{m \in M : f(m) = 0_N\}. \end{align*} [/definition] When $R$ is a field, this is the kernel of a linear map between vector spaces. In linear algebra, the kernel is also called the null space, but the word kernel emphasises its functorial and quotient-theoretic role. The algebraic kernel is built from a simpler question that makes sense for any function: which inputs have been sent to a chosen output? Isolating that question gives a neutral language for comparing ordinary level sets, empty preimages, and the special zero-level set that becomes a kernel when algebraic structure is present. [definition: Fibre of a Function] Let $f: X \to Y$ be a function, and let $y_0 \in Y$. The fibre of $f$ over $y_0$ is \begin{align*} f^{-1}(\{y_0\}) = \{x \in X : f(x) = y_0\}. \end{align*} [/definition] In algebraic contexts where $Y$ has a distinguished neutral element $e_Y$, the kernel of a structure-preserving map $f: X \to Y$ is the fibre over $e_Y$. This fibre viewpoint is helpful, but it should not hide the algebra. A fibre over a point outside the image is empty; a nonempty fibre of a group or module homomorphism is a coset of the kernel, while the kernel itself is the fibre that carries a closed algebraic structure. ## Equivalent Characterisations The first practical reason to care about kernels is that they detect injectivity. A homomorphism is injective precisely when its kernel contains only the neutral element. Thus the kernel is a compact certificate for whether the map has collapsed distinct elements. [quotetheorem:792] The condition is necessary because every homomorphism must send the identity to the identity, so the identity can never be removed from the kernel. It is sufficient because any collapse $g_1 \mapsto g_2$ can be translated back to an element killed by the homomorphism. For example, the reduction map $\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z}$ is not injective precisely because all even integers land in the identity class. The theorem is special to homomorphisms: an arbitrary function may have a singleton fibre over a chosen point and still fail to be injective elsewhere. Modules and vector spaces use additive notation and also require compatibility with scalar multiplication, so the same idea needs a version phrased in terms of the zero submodule. This is the injectivity test used when studying linear transformations, exact sequences, and quotient modules. [quotetheorem:7861] Here the obstruction is a whole submodule, not just an isolated element. If a linear map kills a nonzero vector, it also kills every scalar multiple of that vector, so failure of injectivity is automatically structured. This makes the criterion useful in computations: checking whether the nullspace is zero is exactly checking whether the map loses information. The next setting keeps the additive kernel test but asks for compatibility with multiplication as well. For ring homomorphisms, the same injectivity question needs a version compatible with additive zero and multiplication. This theorem is needed when maps arise from evaluation, reduction, or imposing relations and one wants to know whether the representation is faithful. [quotetheorem:7859] These criteria explain why kernels are more useful than merely knowing that a map is not injective. The kernel identifies the exact obstruction to injectivity: elements in the same fibre differ by something the homomorphism kills. Quotienting by the kernel then collapses precisely those indistinguishable elements and leaves the part of the domain that survives faithfully in the image. This fibre-and-quotient viewpoint will reappear in the examples below, without repeating the [first isomorphism theorem](/theorems/791) in each algebraic setting. ## Standard Examples Concrete kernels show that the definition is not just terminology. They compute the exact relation being imposed by a homomorphism. [example: Parity Homomorphism] Consider the group homomorphism $\varphi: \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z}$ defined by $\varphi(n)=\bar n$. By the definition of the kernel of a group homomorphism, an integer $n$ lies in $\ker \varphi$ exactly when $\bar n=\bar 0$ in $\mathbb{Z}/2\mathbb{Z}$. The equality $\bar n=\bar 0$ means that $n-0$ is divisible by $2$, so there is an integer $k$ with $n=2k$. Hence \begin{align*} \ker \varphi=\{n\in\mathbb{Z}:\bar n=\bar 0\}. \end{align*} Equivalently, \begin{align*} \ker \varphi=\{n\in\mathbb{Z}:2\mid n\}. \end{align*} Since $2\mid n$ holds exactly when $n=2k$ for some $k\in\mathbb{Z}$, we get \begin{align*} \ker \varphi=\{2k:k\in\mathbb{Z}\}=2\mathbb{Z}. \end{align*} Thus reduction modulo $2$ sends precisely the even integers to the identity class $\bar 0$, while odd integers map to the other class $\bar 1$. [/example] The preceding example is the prototype for quotient algebra: the map remembers only parity, and the forgotten information is the subgroup $2\mathbb{Z}$. A linear map can have a kernel with genuine geometry. The kernel is the subspace of directions in which the map has no effect. [example: Projection from a Plane] Let $T: \mathbb{R}^2 \to \mathbb{R}$ be defined by $T(x_1,x_2)=x_1$. By the definition of the kernel of a module homomorphism, a vector $(x_1,x_2)$ lies in $\ker T$ exactly when $T(x_1,x_2)=0$. For any $(x_1,x_2)\in \mathbb{R}^2$, the defining formula for $T$ gives \begin{align*} T(x_1,x_2)=0 \iff x_1=0. \end{align*} Thus the vectors killed by $T$ are precisely those whose first coordinate is zero: \begin{align*} \ker T=\{(x_1,x_2)\in \mathbb{R}^2:x_1=0\}. \end{align*} Writing the free second coordinate as $t\in\mathbb{R}$ gives \begin{align*} \ker T=\{(0,t):t\in\mathbb{R}\}. \end{align*} Equivalently, \begin{align*} \ker T=\{t(0,1):t\in\mathbb{R}\}=\operatorname{span}\{(0,1)\}. \end{align*} So $T$ forgets the second coordinate, and its kernel is the vertical axis, a one-dimensional subspace of $\mathbb{R}^2$. [/example] This example also explains why kernels are central to solving linear equations. The set of solutions to $T(x)=b$ is either empty or an affine translate of $\ker T$. Ring kernels demonstrate why ideals are the right objects for quotient rings. Evaluation maps are especially useful because their kernels encode vanishing conditions. [example: Evaluation Homomorphism] Let $k$ be a field and let $\operatorname{ev}_0: k[x] \to k$ be the ring homomorphism defined by $\operatorname{ev}_0(p)=p(0)$. We compute its kernel from the coefficients of an arbitrary polynomial \begin{align*} p(x)=a_0+a_1x+\cdots+a_nx^n \in k[x]. \end{align*} Evaluating at $0$ gives \begin{align*} p(0)=a_0+a_1\cdot 0+\cdots+a_n\cdot 0^n=a_0. \end{align*} Therefore $p \in \ker \operatorname{ev}_0$ exactly when $p(0)=0$, and this is exactly when $a_0=0$. If $a_0=0$, then \begin{align*} p(x)=a_1x+a_2x^2+\cdots+a_nx^n. \end{align*} Factoring one copy of $x$ from every term gives \begin{align*} p(x)=x(a_1+a_2x+\cdots+a_nx^{n-1}). \end{align*} Thus $p \in (x)$. Conversely, if $p \in (x)$, then $p(x)=xq(x)$ for some $q(x)\in k[x]$, so \begin{align*} p(0)=0\cdot q(0)=0. \end{align*} Hence $p \in \ker \operatorname{ev}_0$. The two inclusions give \begin{align*} \ker \operatorname{ev}_0=(x). \end{align*} Thus the kernel is precisely the ideal of polynomials whose constant term is zero, equivalently the polynomials that vanish at $0$. [/example] The same idea underlies algebraic geometry: functions vanishing on a set form ideals, and quotient rings identify functions that agree on the space being studied. A kernel can be large without making the homomorphism useless. It may be exactly the relation one intended to impose. [example: Sign Homomorphism] Let $S_n$ be the symmetric group and let $\operatorname{sgn}: S_n \to \{1,-1\}$ be the [sign homomorphism](/theorems/778), where $\operatorname{sgn}(\sigma)=1$ when $\sigma$ is even and $\operatorname{sgn}(\sigma)=-1$ when $\sigma$ is odd. The identity element of the codomain group $\{1,-1\}$ under multiplication is $1$, so by the definition of the kernel of a group homomorphism, \begin{align*} \ker(\operatorname{sgn})=\{\sigma\in S_n:\operatorname{sgn}(\sigma)=1\}. \end{align*} By the defining property of the sign map, \begin{align*} \operatorname{sgn}(\sigma)=1 \iff \sigma \text{ is an even permutation}. \end{align*} Therefore \begin{align*} \ker(\operatorname{sgn})=\{\sigma\in S_n:\sigma \text{ is an even permutation}\}. \end{align*} The alternating group is defined as the subgroup of even permutations, \begin{align*} A_n=\{\sigma\in S_n:\sigma \text{ is an even permutation}\}. \end{align*} Hence the two sets have the same elements, and \begin{align*} \ker(\operatorname{sgn})=A_n. \end{align*} Thus the sign homomorphism kills exactly the even permutations, so its kernel is the naturally occurring normal subgroup $A_n$ of $S_n$. [/example] ## Closure and Quotient Structure ### Closure Conditions The kernel is not an arbitrary subset. Its closure under the relevant operations is the main reason it can be used to form quotients. The precise closure condition depends on the ambient algebraic structure: groups require normal subgroups, rings require ideals, and modules require submodules. [quotetheorem:791] Normality is exactly the condition required for the [quotient group](/theorems/790) $G / \ker \varphi$ to inherit a group structure. In a ring, quotienting has an additional obstruction: multiplication by arbitrary ring elements must not move a killed element outside the killed set. That is why the kernel must be an ideal, not merely an additive subgroup. [quotetheorem:851] The ideal property says that multiplying an element killed by $\varphi$ by any element of $R$ still produces an element killed by $\varphi$. This is precisely what makes multiplication well-defined modulo the kernel. For example, if a ring homomorphism sends a polynomial to its value at a point, then two polynomials with the same value should remain equivalent after being multiplied by any third polynomial. The ideal condition is the mechanism that makes this compatibility true. Without it, addition alone might behave well while multiplication on residue classes would depend on the chosen representative. For modules and vector spaces, quotienting has its own closure requirement: the elements killed by the map must remain killed after addition and after multiplication by scalars. The next structural issue is therefore whether the kernel of a module homomorphism automatically has this closure, so that it can serve as a legitimate submodule for forming quotients. [quotetheorem:862] This theorem is the conceptual form of the homogeneous solution principle from linear algebra. If two elements are killed by a linear map, then every linear combination of them is also killed. It also explains why solution spaces of homogeneous linear systems are subspaces rather than just sets of unrelated solutions. The hypothesis that the map respects scalar multiplication is essential: an arbitrary function can have a fiber over zero with no linear structure at all. ### Quotients by Kernels These closure results make quotienting by a kernel legal. The next question is what quotient they produce. For groups, the first isomorphism theorem identifies the quotient by the kernel with the part of the target actually reached by the homomorphism, turning the informal idea of "collapsing exactly what the map kills" into a precise structural statement. [quotetheorem:842] This result explains the slogan that the image is the domain modulo what the map kills. It should be read as a structural identification, not merely as a way to count elements: the quotient inherits the same operation pattern as the image because the homomorphism respects the operations before passing to the quotient. The same question has to be asked again for rings, where addition and multiplication must both survive passage to a quotient. The ring first isomorphism theorem answers this by requiring the kernel to be an ideal: after elements of the kernel are declared zero, multiplication remains well defined and the quotient ring carries exactly the ring structure visible in the image. This is the mechanism behind presentations of rings by generators and relations, where the relations form the kernel of a map from a polynomial or free algebraic object. ### Numerical Size Modules have the same quotient-by-kernel principle, now with scalar multiplication preserved. The module first isomorphism theorem is needed for exact sequences and homological algebra, and it contains the vector-space version used throughout linear algebra. In finite dimensions, the structural statement can be turned into a numerical one: dimension measures how large the kernel is and how much of the domain survives in the image. [quotetheorem:385] Rank-nullity makes the size of a kernel numerically visible, but only in the finite-dimensional vector-space setting where dimension is available and additive in the required way. For a linear map $T:V \to W$, the formula says that solving the homogeneous equation $T(v)=0$ and measuring the span of the outputs are complementary tasks: extra independent solutions in the kernel reduce the possible rank by the same amount. For example, a projection from a plane onto a line has a one-dimensional kernel and a one-dimensional image. The theorem is therefore the bridge from the structural language of kernels and images to computations with systems of linear equations, bases, and exact sequences. ## Relationship to Other Concepts Kernels should be read together with images. The image records what a homomorphism reaches, while the kernel records what it kills. The two are joined by the first isomorphism theorem, which is why image, kernel, and quotient form a single package in algebra. A normal subgroup is exactly the kind of subgroup that can occur as the kernel of a group homomorphism. To reverse the earlier normality theorem, we need the quotient homomorphism attached to a normal subgroup. The following theorem shows that normality is precisely the condition for being killed by some quotient map. [quotetheorem:7834] This theorem justifies thinking of a normal subgroup as a relation that can be imposed by a homomorphism. The ring version is analogous but should be kept at the ordinary algebraic level: an ideal is exactly the kind of additive subgroup of a ring that can be declared zero without destroying addition and multiplication. If $I$ is an ideal of a ring $R$, the quotient map $R \to R/I$ has kernel $I$. Conversely, the kernel of any ring homomorphism is an ideal. Thus ideals play for quotient rings the same role that normal subgroups play for quotient groups. With the convention that quotient homomorphisms land in unital rings with $0 \ne 1$, the ideal must be proper; if the zero ring is admitted as a unital target, the whole ring also appears as the kernel of the map $R \to R/R$. The module case needs the same reversal because quotient modules are built by declaring a chosen submodule to be zero. To use kernels as the basic language for module quotients and exact sequences, every submodule must arise from an actual homomorphism, not merely satisfy a similar closure condition. The next theorem supplies that realization. [quotetheorem:7860] Once every submodule can be realised as a kernel, kernels become a way to track how algebraic information moves through several maps in succession. The next question is no longer just "what does this map kill?", but "is everything killed by this map exactly what arrived from the previous one?" Exactness is the language that records this match between an image and the next kernel. It is needed because long algebraic constructions often contain many maps, and the essential question is whether each stage loses exactly the information supplied by the preceding stage. [definition: Exactness at a Term] Let \begin{align*} A \xrightarrow{f} B \xrightarrow{g} C \end{align*} be homomorphisms of algebraic objects for which kernels and images are defined. The sequence is exact at $B$ if \begin{align*} \operatorname{im} f = \ker g. \end{align*} [/definition] Exactness says that everything killed by the next map came from the previous map. In homological algebra, this principle is organised into long chains of groups or modules, and kernels become the basic language for measuring failure of exactness. [remark: Kernel Versus Null Space] In linear algebra, the terms kernel and null space often refer to the same subspace. The term null space stresses solving $T(x)=0$, while kernel stresses the role of $T$ as a homomorphism and prepares the connection to quotients, exact sequences, and universal constructions. [/remark] The word kernel also appears outside algebra, such as kernels of integral operators or reproducing kernels in analysis. Those notions share the idea of something central to an operator or representation, but they are different definitions. On this page, kernel always means the algebraic preimage of the neutral element under a structure-preserving map. ## Beyond and Connections Kernels are the entry point to several later ideas. The [Quotient Group](/theorems/790) packages the operation of identifying elements that differ by something in the kernel, while a [Normal Subgroup](/page/Normal%20Subgroup) is exactly the kind of subgroup that can occur as the kernel of a group homomorphism. In linear algebra and module theory, the same role is played by the quotient module and by exactness: a sequence is exact at a term when the image entering that term equals the kernel leaving it. For rings, kernels are ideals, so quotient rings record the effect of imposing equations such as $x=0$ or reducing integers modulo $n$. The first isomorphism theorem is the bridge among these viewpoints. It says that once the kernel has been collapsed, the remaining information in the domain is precisely the image of the homomorphism. This is why kernels appear repeatedly in algebraic constructions: they are not only sets of elements sent to zero, but the algebraic instructions for what must be identified before a map becomes faithful. ## References Dummit and Foote, *Abstract Algebra* (2004). Lang, *Algebra* (2002). Atiyah and Macdonald, *Introduction to Commutative Algebra* (1969). [Group](/page/Group), [Ring](/page/Ring), [Module](/page/Module), [Linear Map](/page/Linear%20Map), [Quotient Group](/theorems/790).