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.
Conversely, one can show a bijective module homomorphism is an isomorphism; i.e., the inverse is a module homomorphism.
The isomorphism theorems hold for module homomorphisms.
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 the language of the category theory, an injective homomorphism is also called a monomorphism and a surjective homomorphism an epimorphism.
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 a ring action since Note: the above verification would "fail" if one used the left R-action in place of the right S-action.
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.
, 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.
In practice, one often defines a module homomorphism by specifying its values on a generating set.
Suppose a subset S generates M; i.e., there is a surjection
are module homomorphisms, then their direct sum is and their tensor product is Let
The transpose of f is 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.,
A chain complex is called an exact sequence if
means the localization at a maximal ideal
be commutative rings, and let I be the annihilator of the quotient B-module A/B (which is an ideal of A).
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.)
[3] In other words, it is a "many-valued" homomorphism defined on some submodule of M. The inverse
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.