Wikipedia / Module homomorphism
Wikipedia / Module homomorphism
In algebra, a module homomorphism is a function between moduless that preserves the module structures. Explicitly, if M and N are left modules over a ring R 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. 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 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
is the annihilator of I.
Given a ring R and an element r denote the left multiplication by r. Then for any s, t in R,
.
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.
as rings when R is viewed as a left module over itself. Textbooks or other references usually specify which convention is used.
for any left module M. (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.
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, .
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,
is R-linear) since
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 is a right S.
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
consisting of column vectors and then writing f as an m × n matrix. In particular, viewing R as a right R, 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. 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 with a free module F with a basis indexed by S and kernel K (i.e., one has a free presentation that kills K (i.e., maps K to zero).
Operations
are module homomorphisms, then their direct sum is
and their tensor product is
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
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. (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. A special case of an exact sequence is a short exact sequence:
is surjective.
defines an exact sequence
is the cokernel.
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
means the localization.
are module homomorphisms, then they are said to form a fiber square (or pullback square), denoted by M ×B N, if it fits into
.
be commutative rings, and let I be the annihilator of the quotient B-module A/B (which is an ideal of A
Endomorphisms of finitely generated modules
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.
is surjective, then it is injective.
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 In other words, it is a "many-valued" homomorphism defined on some submodule of M of f. Any additive relation f determines a homomorphism from a submodule of M to a quotient of N
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