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 ringR, 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.
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.
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.
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 representationEdit
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
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.
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 subsetS 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).
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 Γf 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.
If f is an isomorphism, then the transpose of the inverse of f is called the contragredient of f.
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 .
See also: Herbrand quotient (which can be defined for any endomorphism with some finiteness conditions.)
Variant: additive relationsEdit
An additive relation from a module M to a module N is a submodule of  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.