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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.

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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.

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## Definition

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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.

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[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]

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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.

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[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]

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## Algebraic Forms

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### Group Homomorphisms

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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.

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[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]

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### Ring Homomorphisms

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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.

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[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]

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### Module Homomorphisms

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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.

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[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]

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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.

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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.

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[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]

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