# Module homomorphism

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

${\displaystyle f(x+y)=f(x)+f(y),}$
${\displaystyle f(rx)=rf(x).}$

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

${\displaystyle f(xr)=f(x)r.}$

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 ${\displaystyle \operatorname {Hom} _{R}(M,N)}$. 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, 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. One writes ${\displaystyle \operatorname {End} _{R}(M)=\operatorname {Hom} _{R}(M,M)}$ 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, 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 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 MN that maps every element to zero.
• A linear transformation between vector spaces.
• ${\displaystyle \operatorname {Hom} _{\mathbb {Z} }(\mathbb {Z} /n,\mathbb {Z} /m)=\mathbb {Z} /\operatorname {gcd} (n,m)}$.
• For a commutative ring R and ideals I, J, there is the canonical identification
${\displaystyle \operatorname {Hom} _{R}(R/I,R/J)=\{r\in R|rI\subset J\}/J}$
given by ${\displaystyle f\mapsto f(1)}$. In particular, ${\displaystyle \operatorname {Hom} _{R}(R/I,R)}$ is the annihilator of I.
• Given a ring R and an element r, let ${\displaystyle l_{r}:R\to R}$ denote the left multiplication by r. Then for any s, t in R,
${\displaystyle l_{r}(st)=rst=l_{r}(s)t}$.
That is, ${\displaystyle l_{r}}$ is right R-linear.
• For any ring R,
• ${\displaystyle \operatorname {End} _{R}(R)=R}$ as rings when R is viewed as a right module over itself. Explicitly, this isomorphism is given by the left regular representation ${\displaystyle R{\overset {\sim }{\to }}\operatorname {End} _{R}(R),\,r\mapsto l_{r}}$.
• ${\displaystyle \operatorname {Hom} _{R}(R,M)=M}$ through ${\displaystyle f\mapsto f(1)}$ 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.)
• ${\displaystyle \operatorname {Hom} _{R}(M,R)}$ 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 ${\displaystyle M^{*}}$.
• 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

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

${\displaystyle \operatorname {Hom} _{R}(M,N)}$

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

${\displaystyle (s\cdot f)(x)=f(xs).}$

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

${\displaystyle (s\cdot f)(rx)=f(rxs)=rf(xs)=r(s\cdot f)(x),}$

and ${\displaystyle s\cdot f}$ is a ring action since

${\displaystyle (st\cdot f)(x)=f(xst)=(t\cdot f)(xs)=s\cdot (t\cdot f)(x)}$.

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 ${\displaystyle \operatorname {Hom} _{R}(M,N)}$ is a right S-module by ${\displaystyle (f\cdot s)(x)=f(x)s}$.

## 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

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

obtained by viewing ${\displaystyle U^{\oplus n}}$ consisting of column vectors and then writing f as an m × n matrix. In particular, viewing R as a right R-module and using ${\displaystyle \operatorname {End} _{R}(R)\simeq R}$, one has

${\displaystyle \operatorname {End} _{R}(R^{n})\simeq M_{n}(R)}$,

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

## Operations

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

${\displaystyle f\oplus g:M\oplus M'\to N\oplus N',\,(x,y)\mapsto (f(x),g(y))}$

and their tensor product is

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

Let ${\displaystyle f:M\to N}$ be a module homomorphism between left modules. The graph Γf of f is the submodule of MN given by

${\displaystyle \Gamma _{f}=\{(x,f(x))|x\in M\}}$,

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

The transpose of f is

${\displaystyle f^{*}:N^{*}\to M^{*},\,f^{*}(\alpha )=\alpha \circ f.}$

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

## Exact sequences

Consider a sequence of module homomorphisms

${\displaystyle \cdots {\overset {f_{3}}{\longrightarrow }}M_{2}{\overset {f_{2}}{\longrightarrow }}M_{1}{\overset {f_{1}}{\longrightarrow }}M_{0}{\overset {f_{0}}{\longrightarrow }}M_{-1}{\overset {f_{-1}}{\longrightarrow }}\cdots .}$

Such a sequence is called a chain complex (or often just complex) if each composition is zero; i.e., ${\displaystyle f_{i}\circ f_{i+1}=0}$ or equivalently the image of ${\displaystyle f_{i+1}}$ is contained in the kernel of ${\displaystyle f_{i}}$. (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 if ${\displaystyle \operatorname {im} (f_{i+1})=\operatorname {ker} (f_{i})}$. A special case of an exact sequence is a short exact sequence:

${\displaystyle 0\to A{\overset {f}{\to }}B{\overset {g}{\to }}C\to 0}$

where ${\displaystyle f}$ is injective, the kernel of ${\displaystyle g}$ is the image of ${\displaystyle f}$ and ${\displaystyle g}$ is surjective.

Any module homomorphism ${\displaystyle f:M\to N}$ defines an exact sequence

${\displaystyle 0\to K\to M{\overset {f}{\to }}N\to C\to 0,}$

where ${\displaystyle K}$ is the kernel of ${\displaystyle f}$, and ${\displaystyle C}$ is the cokernel, that is the quotient of ${\displaystyle N}$ by the image of ${\displaystyle f}$.

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

${\displaystyle 0\to A_{\mathfrak {m}}{\overset {f}{\to }}B_{\mathfrak {m}}{\overset {g}{\to }}C_{\mathfrak {m}}\to 0}$

are exact, where the subscript ${\displaystyle {\mathfrak {m}}}$ means the localization at a maximal ideal ${\displaystyle {\mathfrak {m}}}$.

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

${\displaystyle 0\to M\times _{B}N\to M\times N{\overset {\phi }{\to }}B\to 0}$

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

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

## Endomorphisms of finitely generated modules

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

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

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

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

${\displaystyle D(f)\to N/\{y|(0,y)\in f\}}$

where ${\displaystyle D(f)}$ 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.