# Kernel (algebra)

In algebra, the **kernel** of a homomorphism f: M → N between two modules over a ring R is the set of all elements of M that f sends to zero: \( \ker(f) = \{ m \in M : f(m) = 0 \} \). It is a submodule of M, and it measures exactly how far f is from being injective.<sup>[1](https://alphabetamath.cc/abstract-algebra/modules-and-module-homomorphisms)</sup><sup> • </sup><sup>[2](https://www.math.ru.nl/personal/bmoonen/CatHomAlg/NotesHomAlgebra.pdf)</sup> A module is an abelian group on which a ring acts by scalar multiplication, so the kernel generalizes the kernel of a linear map between vector spaces, which is the special case where R is a field.<sup>[3](https://jeremy9959.net/Math-5211/01-modules.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( \ker(f) = \{ m \in M : f(m) = 0 \} \) for a module homomorphism f: M → N; it is a submodule of M<sup>[1](https://alphabetamath.cc/abstract-algebra/modules-and-module-homomorphisms)</sup><sup> • </sup><sup>[2](https://www.math.ru.nl/personal/bmoonen/CatHomAlg/NotesHomAlgebra.pdf)</sup> |
| Injectivity test | A module homomorphism is injective exactly when its kernel is trivial<sup>[1](https://alphabetamath.cc/abstract-algebra/modules-and-module-homomorphisms)</sup> |
| Fibers | The kernel is the fiber over zero, and every other fiber of f is a translate of it; each coset of the kernel collapses to a single point of the image<sup>[4](https://notes.amittai.studio/abstract-algebra/module-theory/introduction-to-modules)</sup> |
| Quotients | Every submodule N is the kernel of the projection π: M → M/N, π(x) = x + N, so every submodule arises as a kernel<sup>[4](https://notes.amittai.studio/abstract-algebra/module-theory/introduction-to-modules)</sup><sup> • </sup><sup>[3](https://jeremy9959.net/Math-5211/01-modules.html)</sup> |
| Exact sequences | A sequence of modules and homomorphisms is exact at M_i when im f_{i-1} = ker f_i<sup>[5](https://crypto.stanford.edu/pbc/notes/commalg/exact.html)</sup> |

## Definition and basic properties

Let R be a ring and let f: M → N be a homomorphism of R-modules, meaning an additive map compatible with scalar multiplication. The kernel of f is the submodule \( \ker(f) = \{ m \in M : f(m) = 0 \} \) of M, and the image of f is a submodule of N.<sup>[1](https://alphabetamath.cc/abstract-algebra/modules-and-module-homomorphisms)</sup><sup> • </sup><sup>[2](https://www.math.ru.nl/personal/bmoonen/CatHomAlg/NotesHomAlgebra.pdf)</sup> Both facts follow from the homomorphism properties: the kernel is closed under addition and under scalar multiplication because f(r m) = r f(m) = 0 whenever f(m) = 0.

The kernel controls injectivity. A module homomorphism is injective exactly when its kernel is trivial, that is, when the only element mapped to zero is zero itself.<sup>[1](https://alphabetamath.cc/abstract-algebra/modules-and-module-homomorphisms)</sup>

## Fibers and cosets

The kernel describes the entire mapping, not just the preimage of zero. Under a module homomorphism, all elements of any kernel coset map to one and the same point in the image: the preimage of zero is exactly the kernel, and each remaining preimage is obtained from it by translation.<sup>[4](https://notes.amittai.studio/abstract-algebra/module-theory/introduction-to-modules)</sup> Concretely, if f(m) = f(m′), then m − m′ lies in the kernel, so m and m′ belong to the same coset of \( \ker(f) \). Two elements of M therefore have the same image exactly when they differ by an element of the kernel.

## Kernels and quotient modules

Every submodule is the kernel of a projection, so quotients always exist. The module setting is easier than the general one for groups: because a module's underlying group is abelian, any submodule automatically counts as normal, and nothing blocks the construction of a quotient.<sup>[4](https://notes.amittai.studio/abstract-algebra/module-theory/introduction-to-modules)</sup>

**The projection map.** For a submodule N of M, the quotient abelian group M/N becomes an R-module under the action \( r(x + N) = (r x) + N \), which is well defined because N is closed under scalar multiplication. The projection \( \pi: M \to M/N \), \( \pi(x) = x + N \), is then a surjective R-module homomorphism with \( \ker \pi = N \).<sup>[4](https://notes.amittai.studio/abstract-algebra/module-theory/introduction-to-modules)</sup> The elements of M/N are the cosets x + N, the translates of N that partition M, and a ring element acts on a coset by acting on any of its representatives.<sup>[6](https://www.mathreference.com/mod%2Chomo.html)</sup>

**Worked example.** Let M = \( \mathbb{Z}^2 \) and let N be the submodule generated by (2, 0) and (0, 3), that is \( N = \{(2s, 3t) : s, t \in \mathbb{Z}\} \). A pair (a, b) and a pair (a′, b′) fall into a common coset if and only if a ≡ a′ (mod 2) and b ≡ b′ (mod 3); hence the map

\[ \varphi: \mathbb{Z}^2 \to \mathbb{Z}/2 \oplus \mathbb{Z}/3, \quad \varphi(a, b) = (a \bmod 2,\ b \bmod 3) \]

is a surjective \( \mathbb{Z} \)-module homomorphism with kernel N, and \( M/N \cong \mathbb{Z}/2 \oplus \mathbb{Z}/3 \), which has six cosets. The quotient is cyclic: the coset (1, 1) + N has order 6, since n · (1, 1) ∈ N requires both 2 | n and 3 | n, that is, 6 | n. By the [Chinese remainder theorem](https://www.edgepedia.org/chinese-remainder-theorem) \( M/N \cong \mathbb{Z}/6 \).<sup>[4](https://notes.amittai.studio/abstract-algebra/module-theory/introduction-to-modules)</sup>

## The first isomorphism theorem

In module language, \( M/\ker(f) \cong \operatorname{im}(f) \), with the isomorphism sending the coset m + ker(f) to f(m).

## Exact sequences and the cokernel

Kernels organize longer chains of maps. A sequence

\[ \cdots \rightarrow M_{i-1} \xrightarrow{f_{i-1}} M_i \xrightarrow{f_i} M_{i+1} \rightarrow \cdots \]

of R-modules and R-module homomorphisms is exact at \( M_i \) if \( \operatorname{im} f_{i-1} = \ker f_i \).<sup>[5](https://crypto.stanford.edu/pbc/notes/commalg/exact.html)</sup> A short exact sequence has three terms, and exactness there says that the middle module has the preceding module's image as the kernel of the map to the last module; in this situation M is noetherian if and only if the kernel module K and the quotient module Q are noetherian, a restatement of the kernel quotient theorem.<sup>[7](https://book.mathreference.com/acc)</sup>

Any module homomorphism f: M → N defines an exact sequence in which K is the kernel of f and C is the cokernel, the quotient of N by the image of f.<sup>[8](https://en.wiki.beta.math.wmflabs.org/wiki/Module_homomorphism)</sup> The cokernel is thus the dual notion to the kernel: the kernel records what f kills in the domain, while the cokernel records what in the codomain is not hit.<sup>[2](https://www.math.ru.nl/personal/bmoonen/CatHomAlg/NotesHomAlgebra.pdf)</sup>

## Special cases and the field case

The kernel concept specializes across algebra. For a ring homomorphism f: S → R, the kernel is again defined as the preimage of zero.<sup>[9](https://andreabianchi.sites.ku.dk/files/2023/03/HomAlg2021.pdf)</sup>

When R is a field, M and N are vector spaces and an R-module homomorphism is just a linear map, so the kernel is the familiar null space.<sup>[3](https://jeremy9959.net/Math-5211/01-modules.html)</sup> Over a general ring, by contrast, submodules need not be direct summands, so the kernel alone does not determine the map up to isomorphism, and the quotient M/ker(f) can differ structurally from a submodule of M.

## References

1. [Modules, Submodules, Quotient Modules and the Isomorphism Theorems, Alphabeta Math](https://alphabetamath.cc/abstract-algebra/modules-and-module-homomorphisms)
2. [Notes on Homological Algebra, Radboud University Nijmegen](https://www.math.ru.nl/personal/bmoonen/CatHomAlg/NotesHomAlgebra.pdf)
3. [Modules, Abstract Algebra Part II, Jeremy Brazas lecture notes](https://jeremy9959.net/Math-5211/01-modules.html)
4. [Introduction to Modules, Amittai study notes](https://notes.amittai.studio/abstract-algebra/module-theory/introduction-to-modules)
5. [Commutative Algebra: Exact Sequences, Stanford notes](https://crypto.stanford.edu/pbc/notes/commalg/exact.html)
6. [Module Homomorphisms, MathReference](https://www.mathreference.com/mod%2Chomo.html)
7. [Noetherian and Artinian Modules, MathReference](https://book.mathreference.com/acc)
8. [Module homomorphism, testwiki](https://en.wiki.beta.math.wmflabs.org/wiki/Module_homomorphism)
9. [Lecture Notes for Homological Algebra, Andrea Bianchi, University of Copenhagen](https://andreabianchi.sites.ku.dk/files/2023/03/HomAlg2021.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Module theory › Tensor products and bimodules › Module homomorphisms*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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