Module homomorphism

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 formula_1 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 formula_5. 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 formula_6 for the set of all endomorphisms between 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 (a module 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.
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., formula_23 is "R"-linear) since
and formula_23 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 formula_5 is a right "S"-module by formula_28.
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 formula_30 consisting of column vectors and then writing "f" as an "m" × "n" matrix. In particular, viewing "R" as a right "R"-module and using formula_31, 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 formula_33. 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 formula_34 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 formula_35 is to give a module homomorphism formula_36 that kills "K" (i.e., maps "K" to zero).
Operations.
If formula_1 and formula_38 are module homomorphisms, then their direct sum is
and their tensor product is
Let formula_1 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 
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., formula_45 or equivalently the image of formula_46 is contained in the kernel of formula_47. (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 formula_48. A special case of an exact sequence is a short exact sequence:
where formula_50 is injective, the kernel of formula_51 is the image of formula_50 and formula_51 is surjective.
Any module homomorphism formula_54 defines an exact sequence
where formula_56 is the kernel of formula_50, and formula_58 is the cokernel, that is the quotient of formula_59 by the image of formula_50.
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 formula_62 means the localization at a maximal ideal formula_62.
If formula_64 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 formula_66.
Example: Let formula_67 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 formula_68 form a fiber square with formula_69
Endomorphisms of finitely generated modules.
Let formula_70 be an endomorphism between finitely generated "R"-modules for a commutative ring "R". Then
See also: Herbrand quotient (which can be defined for any endomorphism with some finiteness conditions.)
Variant: additive relations.
An additive relation formula_35 from a module "M" to a module "N" is a submodule of formula_74 In other words, it is a "many-valued" homomorphism defined on some submodule of "M". The inverse formula_75 of "f" is the submodule formula_76. Any additive relation "f" determines a homomorphism from a submodule of "M" to a quotient of "N"
where formula_78 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.