Module homomorphism - Alchetron, The Free Social Encyclopedia

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

Terminology

A module homomorphism is called an isomorphism if it admits the inverse homomorphism. A module homomorphism is an isomorphism if and only if it is an isomorphism between the underlying abelian groups. In other words, an inverse function of a module homomorphism, when it exists, is necessary a homomorphism.

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 End R ⁡ ( M ) = Hom R ⁡ ( M , M ) 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; it is called the endomorphism ring of M.

Schur's lemma says that a homomorphism between simple modules (a module having only two submodules) must be either zero or an isomorphism. In particular, the endomorphism ring of a simple module is a division ring.

To use 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.

Hom Z ⁡ ( Z / n , Z / m ) = Z / gcd ⁡ ( n , m ) .

For a commutative ring R and ideals I, J, there is the canonical identification Hom R ⁡ ( R / I , R / J ) = { r ∈ R | r I ⊂ J } / J

given by f ↦ f ( 1 ) . In particular, Hom R ⁡ ( R / I , R ) is the annihilator of I.

Given a ring R and an element r, let l r : R → R denote the left multiplication by r. Then for any s, t in R, l r ( s t ) = r s t = l r ( s ) t .

That is, l r is right R-linear.

For any ring R,

End R ⁡ ( R ) = R as rings when R is viewed as a right module over itself. Explicitly, this isomorphism is given by the left regular representation R → ∼ End R ⁡ ( R ) , r ↦ l r .

Hom R ⁡ ( R , M ) = M through f ↦ f ( 1 ) 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.)

Hom R ⁡ ( M , R ) 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 M ∗ .

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

Hom R ⁡ ( M , N )

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

( s ⋅ f ) ( x ) = f ( x s ) .

It is well-defined (i.e., s ⋅ f is R-linear) since

( s ⋅ f ) ( r x ) = f ( r x s ) = r f ( x s ) = r ( s ⋅ f ) ( x ) .

Similarly, s ⋅ f is a ring action since

( s t ⋅ f ) ( x ) = f ( x s t ) = ( t ⋅ f ) ( x s ) = s ⋅ ( t ⋅ f ) ( x ) .

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 Hom R ⁡ ( M , N ) is a right S-module by ( f ⋅ s ) ( x ) = f ( x ) s .

A matrix representation of a module homomorphism

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

Hom R ⁡ ( U ⊕ n , U ⊕ m ) → ∼ f ↦ [ f i j ] M m , n ( End R ⁡ ( U ) )

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

End R ⁡ ( R n ) ≃ M n ( R ) ,

which turns out to be a ring isomorphism.

Note the above isomorphism is canonical: 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 F ≃ R n . The above procedure then gives the matrix representation with respect to such choices of the bases.

To define a module homomorphism

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

Operations

If f : M → N and g : M ′ → N ′ are module homomorphisms, then their direct sum is

f ⊕ g : M ⊕ M ′ → N ⊕ N ′ , ( x , y ) ↦ ( f ( x ) , g ( y ) )

and their tensor product is

f ⊗ g : M ⊗ M ′ → N ⊗ N ′ , x ⊗ y ↦ f ( x ) ⊗ g ( y ) .

Let f : M → N be a module homomorphism between left modules. The graph Γ_f of f is the submodule of M ⊕ N given by

Γ f = { ( x , f ( x ) ) | x ∈ M } .

It is a module since it is the image of the graph morphism M → M ⊕ N, x → (x, f(x)).

The transpose of f is

f ∗ : N ∗ → M ∗ , f ∗ ( α ) = α ∘ f .

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

Exact sequences

A short sequence of modules over a commutative ring

0 → A → f B → g C → 0

consists of modules A, B, C and homomorphisms f, g. It is exact if f is injective, the kernel of g is the image of f and g is surjective. A longer exact sequence is defined in the similar way. A sequence of modules is exact if and only if it is exact as a sequence of abelian groups. Also the sequence is exact if and only if it is exact at all the maximal ideals:

0 → A m → f B m → g C m → 0

where the subscript m means the localization of a module at m .

Any module homomorphism f fits into

0 → K → M → f N → C → 0

where K is the kernel of f and C is the cokernel, the quotien of N by the image of f.

If f : M → B , g : N → B are module homomorphisms, then they are said to form a fiber square (or pullback square), denoted by M ×_B N, if it fits into

0 → M × B N → M × N → ϕ B → 0

where ϕ ( x , y ) = f ( x ) − g ( x ) .

Example: Let B ⊂ A 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 A → A / I , B / I → A / I form a fiber square with B = A × A / I B / I .

Endomorphisms of finitely generated modules

Let ϕ : M → M be an endomorphism between finitely generated R-modules for a commutative ring R. Then

ϕ is killed by its characteristic polynomial relative to the generators of M; see Nakayama's lemma#Proof.

If ϕ is surjective, then it is injective.

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

Additive relations

An additive relation M → N from a module M to a module N is a submodule of M ⊕ N . In other words, it is a "many-valued" homomorphism defined on some submodule of M. The inverse f − 1 of f is the submodule { ( y , x ) | ( x , y ) ∈ f } . Any additive relation f determines a homomorphism from a submodule of M to a quotient of N

D ( f ) → N / { y | ( 0 , y ) ∈ f }

where D ( f ) 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.

References

Module homomorphism Wikipedia

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