In algebra, a **module homomorphism** is a function between modules that preserves the module structures. Explicitly, if *M* and *N* are left modules over a ring *R*, then a function is called an *R*-*module homomorphism* or an *R*-*linear map* if for any *x*, *y* in *M* and *r* in *R*,

In other words, *f* is a group homomorphism (for the underlying additive groups) that commutes with scalar multiplication. If *M*, *N* are right *R*-modules, then the second condition is replaced with

The preimage of the zero element under *f* is called the kernel of *f*. The set of all module homomorphisms from *M* to *N* is denoted by . It is an abelian group (under pointwise addition) but is not necessarily a module unless *R* is commutative.

The composition of module homomorphisms is again a module homomorphism, and the identity map on a module is a module homomorphism. Thus, all the (say left) modules together with all the module homomorphisms between them form the category of modules.

## Terminology

A module homomorphism is called a *module isomorphism* if it admits an inverse homomorphism; in particular, it is a bijection. Conversely, one can show a bijective module homomorphism is an isomorphism; i.e., the inverse is a module homomorphism. In particular, a module homomorphism is an isomorphism if and only if it is an isomorphism between the underlying abelian groups.

The isomorphism theorems hold for module homomorphisms.

A module homomorphism from a module *M* to itself is called an endomorphism and an isomorphism from *M* to itself an automorphism. One writes for the set of all endomorphisms of a module *M*. It is not only an abelian group but is also a ring with multiplication given by function composition, called the endomorphism ring of *M*. The group of units of this ring is the automorphism group of *M*.

Schur's lemma says that a homomorphism between simple modules (modules with no non-trivial submodules) must be either zero or an isomorphism. In particular, the endomorphism ring of a simple module is a division ring.

In the language of the category theory, an injective homomorphism is also called a monomorphism and a surjective homomorphism an epimorphism.

## Examples

- The zero map
*M*→*N*that maps every element to zero. - A linear transformation between vector spaces.
- .
- For a commutative ring
*R*and ideals*I*,*J*, there is the canonical identification

- given by . In particular, is the annihilator of
*I*.

- Given a ring
*R*and an element*r*, let denote the left multiplication by*r*. Then for any*s*,*t*in*R*,- .

- That is, is
*right**R*-linear.

- For any ring
*R*,- as rings when
*R*is viewed as a right module over itself. Explicitly, this isomorphism is given by the left regular representation . - Similarly, as rings when
*R*is viewed as a left module over itself. Textbooks or other references usually specify which convention is used. - through for any left module
*M*.^{[1]}(The module structure on Hom here comes from the right*R*-action on*R*; see #Module structures on Hom below.) - is called the dual module of
*M*; it is a left (resp. right) module if*M*is a right (resp. left) module over*R*with the module structure coming from the*R*-action on*R*. It is denoted by .

- as rings when
- Given a ring homomorphism
*R*→*S*of commutative rings and an*S*-module*M*, an*R*-linear map θ:*S*→*M*is called a derivation if for any*f*,*g*in*S*, θ(*f g*) =*f*θ(*g*) + θ(*f*)*g*. - If
*S*,*T*are unital associative algebras over a ring*R*, then an algebra homomorphism from*S*to*T*is a ring homomorphism that is also an*R*-module homomorphism.

## Module structures on Hom

In short, Hom inherits a ring action that was not *used up* to form Hom. More precise, let *M*, *N* be left *R*-modules. Suppose *M* has a right action of a ring *S* that commutes with the *R*-action; i.e., *M* is an (*R*, *S*)-module. Then

has the structure of a left *S*-module defined by: for *s* in *S* and *x* in *M*,

It is well-defined (i.e., is *R*-linear) since

and is a ring action since

- .

Note: the above verification would "fail" if one used the left *R*-action in place of the right *S*-action. In this sense, Hom is often said to "use up" the *R*-action.

Similarly, if *M* is a left *R*-module and *N* is an (*R*, *S*)-module, then is a right *S*-module by .

## A matrix representation

The relationship between matrices and linear transformations in linear algebra generalizes in a natural way to module homomorphisms between free modules. Precisely, given a right *R*-module *U*, there is the canonical isomorphism of the abelian groups

obtained by viewing consisting of column vectors and then writing *f* as an *m* × *n* matrix. In particular, viewing *R* as a right *R*-module and using , one has

- ,

which turns out to be a ring isomorphism (as a composition corresponds to a matrix multiplication).

Note the above isomorphism is canonical; no choice is involved. On the other hand, if one is given a module homomorphism between finite-rank free modules, then a choice of an ordered basis corresponds to a choice of an isomorphism . The above procedure then gives the matrix representation with respect to such choices of the bases. For more general modules, matrix representations may either lack uniqueness or not exist.

## Defining

In practice, one often defines a module homomorphism by specifying its values on a generating set. More precisely, let *M* and *N* be left *R*-modules. Suppose a subset *S* generates *M*; i.e., there is a surjection with a free module *F* with a basis indexed by *S* and kernel *K* (i.e., one has a free presentation). Then to give a module homomorphism is to give a module homomorphism that kills *K* (i.e., maps *K* to zero).

## Operations

If and are module homomorphisms, then their direct sum is

and their tensor product is

Let be a module homomorphism between left modules. The graph Γ_{f} of *f* is the submodule of *M* ⊕ *N* given by

- ,

which is the image of the module homomorphism *M* → *M* ⊕ *N*, *x* → (*x*, *f*(*x*)), called the **graph morphism**.

The transpose of *f* is

If *f* is an isomorphism, then the transpose of the inverse of *f* is called the **contragredient** of *f*.

## Exact sequences

Consider a sequence of module homomorphisms

Such a sequence is called a chain complex (or often just complex) if each composition is zero; i.e., or equivalently the image of is contained in the kernel of . (If the numbers increase instead of decrease, then it is called a cochain complex; e.g., de Rham complex.) A chain complex is called an exact sequence if . A special case of an exact sequence is a short exact sequence:

where is injective, the kernel of is the image of and is surjective.

Any module homomorphism defines an exact sequence

where is the kernel of , and is the cokernel, that is the quotient of by the image of .

In the case of modules over a commutative ring, a sequence is exact if and only if it is exact at all the maximal ideals; that is all sequences

are exact, where the subscript means the localization at a maximal ideal .

If are module homomorphisms, then they are said to form a **fiber square** (or **pullback square**), denoted by *M* ×_{B} *N*, if it fits into

where .

Example: Let be commutative rings, and let *I* be the annihilator of the quotient *B*-module *A*/*B* (which is an ideal of *A*). Then canonical maps form a fiber square with

## Endomorphisms of finitely generated modules

Let be an endomorphism between finitely generated *R*-modules for a commutative ring *R*. Then

- is killed by its characteristic polynomial relative to the generators of
*M*; see Nakayama's lemma#Proof. - If is surjective, then it is injective.
^{[2]}

See also: Herbrand quotient (which can be defined for any endomorphism with some finiteness conditions.)

## Variant: additive relations

An **additive relation** from a module *M* to a module *N* is a submodule of ^{[3]} In other words, it is a "many-valued" homomorphism defined on some submodule of *M*. The inverse of *f* is the submodule . Any additive relation *f* determines a homomorphism from a submodule of *M* to a quotient of *N*

where consists of all elements *x* in *M* such that (*x*, *y*) belongs to *f* for some *y* in *N*.

A transgression that arises from a spectral sequence is an example of an additive relation.

## See also

## Notes

## References

- Bourbaki,
*Algebra*. Chapter II.^{[full citation needed]} - S. MacLane,
*Homology*^{[full citation needed]} - H. Matsumura,
*Commutative ring theory.*Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.