Module homomorphism

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 $f:M\to N$ is called a module homomorphism or an R-linear map if for any x, y in M and r in R,

f(x+y)=f(x)+f(y),

f(rx)=rf(x).

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

f(xr)=f(x)r.

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 Hom_R(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

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 $\operatorname {End} _{R}(M)=\operatorname {Hom} _{R}(M,M)$ 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.
$\operatorname {Hom}_{{{\mathbb {Z}}}}({\mathbb {Z}}/n,{\mathbb {Z}}/m)={\mathbb {Z}}/\operatorname {gcd}(n,m)$ .
For a commutative ring R and ideals I, J, there is the canonical identification
$\operatorname {Hom} _{R}(R/I,R/J)=\{r\in R|rI\subset J\}/J$

given by

f\mapsto f(1)

. In particular,

\operatorname {Hom} _{R}(R/I,R)

is the annihilator of I.

Given a ring R and an element r, let $l_{r}:R\to R$ denote the left multiplication by r. Then for any s, t in R,
$l_{r}(st)=rst=l_{r}(s)t$ .

That is,

l_{r}

is right R-linear.

For any ring R,
- $\operatorname {End}_{R}(R)=R$ as rings when R is viewed as a right module over itself. Explicitly, this isomorphism is given by the left regular representation $R{\overset {\sim }{\to }}\operatorname {End} _{R}(R),\,r\mapsto l_{r}$ .
- $\operatorname {Hom}_{R}(R,M)=M$ through $f\mapsto f(1)$ 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.)
- $\operatorname {Hom}_{R}(M,R)$ 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

\operatorname{Hom}_R(M, N)

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

(s\cdot f)(x)=f(xs).

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

(s\cdot f)(rx)=f(rxs)=rf(xs)=r(s\cdot f)(x).

Similarly, $s\cdot f$ is a ring action since

(st\cdot f)(x)=f(xst)=(t\cdot f)(xs)=s\cdot (t\cdot 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 $\operatorname{Hom}_R(M, N)$ is a right S-module by $(f\cdot s)(x)=f(x)s$ .

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

\operatorname {Hom} _{R}(U^{\oplus n},U^{\oplus m}){\overset {f\mapsto [f_{ij}]}{\underset {\sim }{\to }}}M_{m,n}(\operatorname {End} _{R}(U))

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

\operatorname {End} _{R}(R^{n})\simeq 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 $F\simeq R^{n}$ . 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 $F\to M$ 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 $M\to N$ is to give a module homomorphism $F\to N$ that kills K (i.e., maps K to zero).

Operations

If $f:M\to N$ and $g:M'\to N'$ are module homomorphisms, then their direct sum is

f \oplus g: M \oplus M' \to N \oplus N', \, (x, y) \mapsto (f(x), g(y))

and their tensor product is

f \otimes g: M \otimes M' \to N \otimes N', \, x \otimes y \mapsto f(x) \otimes g(y).

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

\Gamma _{f}=\{(x,f(x))|x\in 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^{*}\to M^{*},\,f^{*}(\alpha )=\alpha \circ 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\to A{\overset {f}\to }B{\overset {g}\to }C\to 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\to A_{{{\mathfrak {m}}}}{\overset {f}\to }B_{{{\mathfrak {m}}}}{\overset {g}\to }C_{{{\mathfrak {m}}}}\to 0

where the subscript ${{\mathfrak {m}}}$ means the localization of a module at ${{\mathfrak {m}}}$ .

Any module homomorphism f fits into

0\to K\to M{\overset {f}\to }N\to C\to 0

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

If $f:M\to B,g:N\to B$ 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\to M\times _{{B}}N\to M\times N{\overset {\phi }\to }B\to 0

where $\phi (x,y)=f(x)-g(x)$ .

Example: Let $B\subset A$ 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 $A\to A/I,B/I\to A/I$ form a fiber square with $B=A\times _{{A/I}}B/I.$

Endomorphisms of finitely generated modules

Let $\phi :M\to M$ be an endomorphism between finitely generated R-modules for a commutative ring R. Then

$\phi$ is killed by its characteristic polynomial relative to the generators of M; see Nakayama's lemma#Proof.
If $\phi$ is surjective, then it is injective.^[2]

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

Variants

Additive relations

An additive relation $M\to N$ from a module M to a module N is a submodule of $M\oplus N.$ ^[3] In other words, it is a "many-valued" homomorphism defined on some submodule of M. The inverse $f^{-1}$ of f is the submodule $\{(y,x)|(x,y)\in f\}$ . Any additive relation f determines a homomorphism from a submodule of M to a quotient of N

D(f)\to N/\{y|(0,y)\in 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.

Notes

↑ Bourbaki, § 1.14
↑ Matsumura, Theorem 2.4.
↑

References

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

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