Module homomorphism

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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 is called a module homomorphism or an R-linear map if for any x, y in M and r in R,

If M, N are right R-modules, then the second condition is replaced with

The pre-image 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 HomR(M, 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. Thus, all the (say left) modules together with all the module homomorphisms between them form the category of modules.

Terminology[edit]

A module homomorphism is called an isomorphism if it admits an inverse homomorphism; in particular, it is a bijection. 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 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 having only two 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[edit]

  • The zero map MN 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
given by . In particular, is the annihilator of I.
  • Given a ring R and an element r, let denote the left multiplication by r. Then for any s, t in R,
    .
That is, 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 .
    • through for any left module M.[1] (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. It is denoted by .
  • Given a ring homomorphism RS of commutative rings and an S-module M, an R-linear map θ: SM 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[edit]

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

Similarly, 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 representation[edit]

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

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

,

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.

Defining[edit]

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 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).

Operations[edit]

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 MN given by

,

which is the image of the module homomorphism MMN, x → (x, f(x)).

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[edit]

A short sequence of modules

consists of modules A, B, C, and homomorphisms f, g. It is exact if the image of any arrow is the kernel of the next one; that is, f is injective, the kernel of g is the image of f and g is surjective. A longer exact sequence is defined in a similar way. A sequence of modules is exact if and only if it is exact as a sequence of abelian groups.

Any module homomorphism f defines an exact sequence

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

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 means the localization at a maximal ideal .

If 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 .

Example: Let 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 form a fiber square with

Endomorphisms of finitely generated modules[edit]

Let 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.
  • If is surjective, then it is injective.[2]

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

Variants[edit]

Additive relations[edit]

An additive relation from a module M to a module N is a submodule of [3] 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.

See also[edit]

Notes[edit]

  1. ^ Bourbaki, § 1.14
  2. ^ Matsumura, Theorem 2.4.
  3. ^ MacLane, Saunders (2012-12-06). Homology. Springer Science & Business Media. ISBN 9783642620294.

References[edit]

  • Bourbaki, Algebra[full citation needed]
  • S. MacLane, Homology[full citation needed]
  • H. Matsumura, Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.