# Module homomorphism

A module homomorphism is a map \( f : M \to N \) between two modules over a ring \( R \) that preserves addition and scalar multiplication: \( f(x + y) = f(x) + f(y) \) and \( f(rx) = r f(x) \).<sup>[1](https://kevos.com/engineering/module-homomorphisms-and-isomorphisms)</sup> Such a map is also called an \( R \)-linear map, and when \( R \) is a field it is precisely a linear map between vector spaces.<sup>[1](https://kevos.com/engineering/module-homomorphisms-and-isomorphisms)</sup><sup> • </sup><sup>[3](https://algebrica.org/modules/)</sup> Module homomorphisms are the structure-preserving maps of module theory, the way group homomorphisms are for groups; a large part of what is known about a ring \( R \) is extracted not just from its modules but from the homomorphisms between them.<sup>[9](https://webspace.maths.qmul.ac.uk/b.jackson/MAS427.html)</sup>

| Key fact | Detail |
|---|---|
| Defining property | \( f(x + y) = f(x) + f(y) \) and \( f(rx) = r f(x) \) for all \( x, y \in M \), \( r \in R \); over a field this is exactly a linear map<sup>[1](https://kevos.com/engineering/module-homomorphisms-and-isomorphisms)</sup> |
| Kernel and image | \( \ker f = \{x \in M : f(x) = 0\} \) is a submodule of \( M \), and \( \operatorname{im} f \) is a submodule of \( N \)<sup>[4](https://docs.sympy.org/latest/modules/polys/agca.html)</sup> |
| First isomorphism theorem | \( M / \ker f \cong \operatorname{im} f \); more generally, any submodule \( K \subseteq \ker f \) induces a well-defined homomorphism \( \bar{f} : M/K \to N \)<sup>[2](https://link.springer.com/chapter/10.1007/978-3-319-19734-0_8)</sup> |
| Injectivity and surjectivity | \( f \) is injective if and only if \( \ker f = 0 \), and surjective if and only if \( \operatorname{im} f = N \); a bijective homomorphism is an isomorphism<sup>[4](https://docs.sympy.org/latest/modules/polys/agca.html)</sup> |
| The Hom module | \( \operatorname{Hom}_R(M, N) \), the set of all homomorphisms \( M \to N \), is itself an \( R \)-module under pointwise operations<sup>[5](https://math.mit.edu/classes/18.721/notes/ag-jan26-2022.pdf)</sup> |
| Free-module case | A homomorphism \( A^n \to A^m \) of free modules is specified by an \( m \times n \) matrix, exactly as for vector spaces<sup>[4](https://docs.sympy.org/latest/modules/polys/agca.html)</sup> |
| Historical origin | Module theory as taught goes back to Emmy Noether's work at the University of Göttingen<sup>[9](https://webspace.maths.qmul.ac.uk/b.jackson/MAS427.html)</sup> |

## Definition and first examples

For left \( R \)-modules \( M \) and \( N \), a homomorphism is a function \( \varphi : M \to N \) that preserves both the additive structure and the action of the ring.<sup>[3](https://algebrica.org/modules/)</sup> Equivalently, it is a homomorphism of the underlying abelian groups that respects the ring action.<sup>[6](https://crypto.stanford.edu/pbc/notes/commalg/module.html)</sup> The two conditions are independent of each other, and the ring condition is the one that carries the extra content: over \( R = F[x] \), the ring map \( f(x) \mapsto f(x^2) \) is a homomorphism of abelian groups that fails to be \( F[x] \)-linear.<sup>[8](https://notes.amittai.studio/abstract-algebra/module-theory/introduction-to-modules)</sup>

**Every module homomorphism is a group homomorphism, but not conversely.** Over \( \mathbb{Z} \), the map \( x \mapsto 2x \) is a \( \mathbb{Z} \)-module homomorphism, since every abelian group is a \( \mathbb{Z} \)-module and any group homomorphism respects multiplication by integers.<sup>[8](https://notes.amittai.studio/abstract-algebra/module-theory/introduction-to-modules)</sup> Over a general ring the two notions separate, which is why the scalar condition \( f(rx) = r f(x) \) must be stated separately.<sup>[1](https://kevos.com/engineering/module-homomorphisms-and-isomorphisms)</sup>

For a right \( R \)-module the definition is the same with scalars acting on the right; a right \( R \)-module homomorphism is the same thing as a left homomorphism over the opposite ring \( R^{\mathrm{op}} \), and when \( R \) is commutative the two notions coincide.<sup>[7](https://www.math.uni-bielefeld.de/~wcrawley/19noncommalg1/NA1-4.pdf)</sup>

## Kernel, image, and the isomorphism theorem

The kernel of \( f \) is the set \( \ker f = \{m \in M : f(m) = 0\} \), the preimage of zero, and the image \( \operatorname{im} f \) is the usual image of \( f \) as a function.<sup>[4](https://docs.sympy.org/latest/modules/polys/agca.html)</sup> Both inherit module structures: the kernel is a submodule of \( M \) and the image is a submodule of \( N \).<sup>[4](https://docs.sympy.org/latest/modules/polys/agca.html)</sup><sup> • </sup><sup>[1](https://kevos.com/engineering/module-homomorphisms-and-isomorphisms)</sup> A composition of module homomorphisms is again a module homomorphism.<sup>[3](https://algebrica.org/modules/)</sup>

The kernel measures injectivity and the image measures surjectivity: \( f \) is injective exactly when \( \ker f = 0 \) and surjective exactly when \( \operatorname{im} f = N \), so a bijective homomorphism, called an isomorphism, is characterized by \( \ker f = 0 \) and \( \operatorname{im} f = N \).<sup>[4](https://docs.sympy.org/latest/modules/polys/agca.html)</sup>

**The Fundamental Theorem of Module Homomorphisms** states the quotient relationship precisely. If \( f : M \to N \) is a homomorphism of right (or left) \( R \)-modules and \( K \subseteq \ker f \) is a submodule of \( M \), then \( \bar{f}(m + K) := f(m) \) is a well-defined module homomorphism \( \bar{f} : M/K \to N \); when \( K = \ker f \), the induced map \( \bar{f} : M/\ker f \to f(M) \) is an isomorphism.<sup>[2](https://link.springer.com/chapter/10.1007/978-3-319-19734-0_8)</sup> The proof that \( \bar{f} \) respects scalars is one line: \( \bar{f}((x + K)r) = f(xr) = f(x)r = (\bar{f}(x + K))r \).<sup>[2](https://link.springer.com/chapter/10.1007/978-3-319-19734-0_8)</sup> Put verbally, the map identifies all elements of any kernel coset, sending each one to one point of the image; the kernel constitutes the preimage of zero, and each remaining fiber is a translate of the kernel.<sup>[8](https://notes.amittai.studio/abstract-algebra/module-theory/introduction-to-modules)</sup>

Two companion isomorphism theorems hold in the same module form: for submodules \( L \) and \( N \) of a module \( M \), \( L/(L \cap N) \cong (L + N)/N \); and the submodules of \( M/N \) are exactly the \( L/N \) for submodules \( L \supseteq N \), with \( (M/N)/(L/N) \cong M/L \).<sup>[7](https://www.math.uni-bielefeld.de/~wcrawley/19noncommalg1/NA1-4.pdf)</sup>

A concrete application shows how quotients by ideals arise. For any element \( m \) of a left \( R \)-module \( M \), the surjective homomorphism \( R \to M \) sending \( r \) to \( rm \) has as kernel the left ideal \( I = \{r : rm = 0\} \), so the first isomorphism theorem gives \( R/I \cong \) the cyclic submodule generated by \( m \).<sup>[12](https://math.ucla.edu/~sharifi/notes/algebra-ch05.html)</sup>

## The module of homomorphisms

The set \( \operatorname{Hom}_R(M, N) \) of all homomorphisms \( M \to N \) carries its own module structure: for homomorphisms \( \varphi \) and \( \psi \) and \( a \in A \), the maps \( \varphi + \psi \) and \( a\varphi \) are defined pointwise by \( [\varphi + \psi](m) = \varphi(m) + \psi(m) \) and \( [a\varphi](m) = a\varphi(m) \).<sup>[5](https://math.mit.edu/classes/18.721/notes/ag-jan26-2022.pdf)</sup> This is the starting point for homological algebra.<sup>[1](https://kevos.com/engineering/module-homomorphisms-and-isomorphisms)</sup> Over a noncommutative ring the module structure on Hom requires care: if \( P \) is an \( A \)-\( B \)-bimodule, then \( \operatorname{Hom}_B(P, N) \) becomes a right \( A \)-module by the rule \( [fa](p) = f(ap) \), and a left \( R \)-module structure arises from a bimodule structure \( {}_R N_B \) by \( [rf](p) = r(f(p) \).<sup>[11](https://ocw.mit.edu/courses/18-706-noncommutative-algebra-spring-2023/mit18_706_s23_notes_by_michael_artin.pdf)</sup>

Hom is also a functor with exactness properties: it is left exact, contravariant in the first variable, and covariant in the second, so an exact sequence \( M_1 \to M_2 \to M_3 \to 0 \) induces an exact sequence \( 0 \to \operatorname{Hom}_A(M_3, N) \to \operatorname{Hom}_A(M_2, N) \to \operatorname{Hom}_A(M_1, N) \).<sup>[5](https://math.mit.edu/classes/18.721/notes/ag-jan26-2022.pdf)</sup> This one-sided exactness is what motivates the definitions of projective and [injective modules](https://www.edgepedia.org/injective-module): \( P \) is projective exactly when \( \operatorname{Hom}(P, -) \) is exact, and \( P \) is injective exactly when \( \operatorname{Hom}(-, P) \) is exact.<sup>[11](https://ocw.mit.edu/courses/18-706-noncommutative-algebra-spring-2023/mit18_706_s23_notes_by_michael_artin.pdf)</sup>

Two small computations anchor the theory. An \( A \)-module \( N \) is canonically isomorphic to \( \operatorname{Hom}_A(A, N) \), the homomorphism corresponding to \( v \in N \) being multiplication by \( v \), and \( v \) being recovered as \( \varphi(1) \).<sup>[5](https://math.mit.edu/classes/18.721/notes/ag-jan26-2022.pdf)</sup> Consequently \( \operatorname{Hom}_A(A^k, N) \cong N^k \) and \( \operatorname{Hom}_A(A^k, A^\ell) \) is the module of \( \ell \times k \) matrices over \( A \).<sup>[5](https://math.mit.edu/classes/18.721/notes/ag-jan26-2022.pdf)</sup> When \( A \) is noetherian and \( M \), \( N \) are finite \( A \)-modules, \( \operatorname{Hom}_A(M, N) \) is again finite.<sup>[5](https://math.mit.edu/classes/18.721/notes/ag-jan26-2022.pdf)</sup>

## Computation: kernels and syzygies

For finitely generated modules, homomorphisms become objects a computer can manipulate. A module \( M \) is finitely generated when there is a surjective homomorphism \( A^n \to M \); the kernel of such a presentation map is called the syzygy (module of relations among the chosen generators) module with respect to the chosen generators, and \( M \) is finitely presented when that syzygy module is also finitely generated.<sup>[4](https://docs.sympy.org/latest/modules/polys/agca.html)</sup> Over the polynomial rings used in computer algebra, all submodules of finitely generated modules are finitely generated, so finitely generated and finitely presented coincide for these rings.<sup>[4](https://docs.sympy.org/latest/modules/polys/agca.html)</sup>

Computing a kernel of a module homomorphism means solving a system of linear equations over the ring, and there are algorithms for it: from an algorithm computing the syzygy module of a [Gröbner basis](https://www.edgepedia.org/grobner-basis) one derives algorithms for kernels of arbitrary matrices, that is, homomorphisms of free modules of finite rank, and from these a fundamental algorithm computing \( \operatorname{Hom}_R(M, N) \) itself.<sup>[13](https://agag-jboehm.math.rptu.de/~boehm/lehre/1213_CA/ca.pdf)</sup> More general questions, such as intersections of submodules, are often reduced to syzygy computation.<sup>[4](https://docs.sympy.org/latest/modules/polys/agca.html)</sup>

## Why the maps matter

The first isomorphism theorem carries a structural implication: any image of \( M \) under a homomorphism arises as a quotient of \( M \), hence classifying \( M \) up to isomorphism comes down to knowing its submodules together with the quotients they determine.<sup>[8](https://notes.amittai.studio/abstract-algebra/module-theory/introduction-to-modules)</sup> This is the reason module theory is organized around submodules, quotient modules, and diagrams of homomorphisms rather than around individual modules in isolation.<sup>[9](https://webspace.maths.qmul.ac.uk/b.jackson/MAS427.html)</sup>

The Hom construction also connects modules to representation theory. A representation of a group \( G \) over a field \( K \) is the same thing as a module over the group ring \( K[G] \), and the \( G \)-homomorphisms between two representations are the \( K[G] \)-module homomorphisms, also called intertwining maps, characterized by \( \varrho_2(g) \circ T = T \circ \varrho_1(g) \) for all \( g \in G \).<sup>[6](https://crypto.stanford.edu/pbc/notes/commalg/module.html)</sup><sup> • </sup><sup>[14](https://mooc.math.aalto.fi/~kkytola/files_KK/Lie2019/Lie_course.pdf)</sup> The historical thread runs through [Emmy Noether](https://www.edgepedia.org/emmy-noether) (1882–1935), the German algebraist whose abstract approach to ring and ideal theory, developed at [Göttingen](https://www.edgepedia.org/gottingen) from 1919 onward, is the origin from which modern module theory courses descend.<sup>[10](https://mathshistory.st-andrews.ac.uk/Biographies/Noether_Emmy/)</sup><sup> • </sup><sup>[9](https://webspace.maths.qmul.ac.uk/b.jackson/MAS427.html)</sup>

## References

1. [Module Homomorphisms and Isomorphisms, Kevin Jogin, kevos.com](https://kevos.com/engineering/module-homomorphisms-and-isomorphisms)
2. [Elementary Properties of Modules, Springer Nature Link (The Fundamental Theorem of Module Homomorphisms)](https://link.springer.com/chapter/10.1007/978-3-319-19734-0_8)
3. [Modules, Algebrica](https://algebrica.org/modules/)
4. [AGCA: Algebraic Geometry and Commutative Algebra Module, SymPy 1.14.0 documentation](https://docs.sympy.org/latest/modules/polys/agca.html)
5. [6.4.16. The module Hom, MIT 18.721 notes](https://math.mit.edu/classes/18.721/notes/ag-jan26-2022.pdf)
6. [Comm. Algebra: Modules, Ben Lynn, Stanford University](https://crypto.stanford.edu/pbc/notes/commalg/module.html)
7. [Noncommutative algebra 1, W. Crawley, Bielefeld](https://www.math.uni-bielefeld.de/~wcrawley/19noncommalg1/NA1-4.pdf)
8. [Introduction to Modules, Amittai study notes](https://notes.amittai.studio/abstract-algebra/module-theory/introduction-to-modules)
9. [Rings and Modules, MAS427, Queen Mary University of London](https://webspace.maths.qmul.ac.uk/b.jackson/MAS427.html)
10. [Emmy Noether (1882–1935), Biography, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Noether_Emmy/)
11. [Noncommutative Rings lecture notes by Prof. Michael Artin, MIT OpenCourseWare 18.706](https://ocw.mit.edu/courses/18-706-noncommutative-algebra-spring-2023/mit18_706_s23_notes_by_michael_artin.pdf)
12. [Abstract Algebra, Chapter 5: Advanced ring theory, Romyar Sharifi, UCLA](https://math.ucla.edu/~sharifi/notes/algebra-ch05.html)
13. [Computer Algebra, J. Böhm, RPTU](https://agag-jboehm.math.rptu.de/~boehm/lehre/1213_CA/ca.pdf)
14. [Basic Theory of Representations of Finite Groups, K. Kytölä, Aalto University](https://mooc.math.aalto.fi/~kkytola/files_KK/Lie2019/Lie_course.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: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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