Inalgebra, amodule homomorphism is afunction betweenmodules that preserves the module structures. Explicitly, ifM andN are left modules over aringR, then a function${\displaystyle f:M\to N}$ is called anR-module homomorphism or anR-linear map if for anyx,y inM andr inR,

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

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

In other words,f is agroup homomorphism (for the underlying additive groups) that commutes withscalar multiplication. IfM,N are rightR-modules, then the second condition is replaced with

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

Thepreimage of the zero element underf is called thekernel off. Theset of all module homomorphisms fromM toN is denoted by${\displaystyle {\mathrm{Hom}}_R(M,N)}$. It is anabelian group (under pointwise addition) but is not necessarily a module unlessR iscommutative.

Thecomposition 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 thecategory of modules.

A module homomorphism is called amodule isomorphism if it admits an inverse homomorphism; in particular, it is abijection. 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 isomorphismif and only if it is an isomorphism between the underlying abelian groups.

Theisomorphism theorems hold for module homomorphisms.

A module homomorphism from a moduleM to itself is called anendomorphism and an isomorphism fromM to itself anautomorphism. One writes${\displaystyle {\mathrm{End}}_R(M)={\mathrm{Hom}}_R(M,M)}$ for the set of all endomorphisms of a moduleM. It is not only an abelian group but is also a ring with multiplication given by function composition, called theendomorphism ring ofM. Thegroup of units of this ring is theautomorphism group ofM.

Schur's lemma says that a homomorphism betweensimple modules (modules with no non-trivialsubmodules) must be either zero or an isomorphism. In particular, the endomorphism ring of a simple module is adivision ring.

In the language of thecategory theory, an injective homomorphism is also called amonomorphism and a surjective homomorphism anepimorphism.

• Thezero mapM →N that maps every element to zero.
• Alinear transformation betweenvector spaces.
• ${\displaystyle {\mathrm{Hom}}_{\mathbb{Z}}(\mathbb{Z}/n,\mathbb{Z}/m)=\mathbb{Z}/\gcd (n,m)}$.
• For a commutative ringR andidealsI,J, there is the canonical identification

  ${\displaystyle {\mathrm{Hom}}_R(R/I,R/J)=\{r\in R|rI\subset J\}/J}$

given by${\displaystyle f\mapsto f(1)}$. In particular,${\displaystyle {\mathrm{Hom}}_R(R/I,R)}$ is theannihilator ofI.

• Given a ringR and an elementr, let${\displaystyle l_r:R\to R}$ denote the left multiplication byr. Then for anys,t inR,

  ${\displaystyle l_r(st)=rst=l_r(s)t}$.

That is,${\displaystyle l_r}$ isrightR-linear.

• For any ringR,
  • ${\displaystyle {\mathrm{End}}_R(R)=R}$ as rings whenR is viewed as a right module over itself. Explicitly, this isomorphism is given by theleft regular representation${\displaystyle R\stackrel{\sim }{\to }{\mathrm{End}}_R(R),r\mapsto l_r}$.
  • Similarly,${\displaystyle {\mathrm{End}}_R(R)=R^{op}}$ as rings whenR is viewed as a left module over itself. Textbooks or other references usually specify which convention is used.
  • ${\displaystyle {\mathrm{Hom}}_R(R,M)=M}$ through${\displaystyle f\mapsto f(1)}$ for any left moduleM.^[1] (The module structure on Hom here comes from the rightR-action onR; see#Module structures on Hom below.)
  • ${\displaystyle {\mathrm{Hom}}_R(M,R)}$ is called thedual module ofM; it is a left (resp. right) module ifM is a right (resp. left) module overR with the module structure coming from theR-action onR. It is denoted by${\displaystyle M^{\ast }}$.
• Given a ring homomorphismR →S of commutative rings and anS-moduleM, anR-linear map θ:S →M is called aderivation if for anyf,g inS,θ(f g) =f θ(g) + θ(f)g.
• IfS,T are unitalassociative algebras over a ringR, then analgebra homomorphism fromS toT is aring homomorphism that is also anR-module homomorphism.

In short, Hom inherits a ring action that was notused up to form Hom. More precise, letM,N be leftR-modules. SupposeM has a right action of a ringS that commutes with theR-action; i.e.,M is an (R,S)-module. Then

${\displaystyle {\mathrm{Hom}}_R(M,N)}$

has the structure of a leftS-module defined by: fors inS andx inM,

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

It is well-defined (i.e.,${\displaystyle s\cdot f}$ isR-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 leftR-action in place of the rightS-action. In this sense, Hom is often said to "use up" theR-action.

Similarly, ifM is a leftR-module andN is an (R,S)-module, then${\displaystyle {\mathrm{Hom}}_R(M,N)}$ is a rightS-module by${\displaystyle (f\cdot s)(x)=f(x)s}$.

The relationship between matrices and linear transformations inlinear algebra generalizes in a natural way to module homomorphisms between free modules. Precisely, given a rightR-moduleU, there is thecanonical isomorphism of the abelian groups

${\displaystyle {\mathrm{Hom}}_R(U^{\oplus n},U^{\oplus m})\stackrel{f\mapsto [f_{ij}]}{\underset{\sim }{\to }}{M}_{m,n}({\mathrm{End}}_R(U))}$

obtained by viewing${\displaystyle U^{\oplus n}}$ consisting of column vectors and then writingf as anm ×n matrix. In particular, viewingR as a rightR-module and using${\displaystyle {\mathrm{End}}_R(R)\simeq R}$, one has

${\displaystyle {\mathrm{End}}_R(R^n)\simeq M_n(R)}$,

which turns out to be a ring isomorphism (as a composition corresponds to amatrix multiplication).

Note the above isomorphism is canonical; no choice is involved. On the other hand, if one is given a module homomorphism between finite-rankfree 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.

In practice, one often defines a module homomorphism by specifying its values on agenerating set. More precisely, letM andN be leftR-modules. Suppose asubsetS generatesM; i.e., there is a surjection${\displaystyle F\to M}$ with a free moduleF with a basis indexed byS and kernelK (i.e., one has afree presentation). Then to give a module homomorphism${\displaystyle M\to N}$ is to give a module homomorphism${\displaystyle F\to N}$ that killsK (i.e., mapsK to zero).

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

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

and their tensor product is

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

Let${\displaystyle f:M\to N}$ be a module homomorphism between left modules. Thegraph Γ_f off is the submodule ofM ⊕N given by

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

which is the image of the module homomorphismM →M ⊕N,x → (x,f(x)), called thegraph morphism.

Thetranspose off is

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

Iff is an isomorphism, then the transpose of the inverse off is called thecontragredient off.

Consider a sequence of module homomorphisms

${\displaystyle \cdots \stackrel{f_3}{⟶}{M}_2\stackrel{f_2}{⟶}{M}_1\stackrel{f_1}{⟶}{M}_0\stackrel{f_0}{⟶}{M}_{-1}\stackrel{f_{-1}}{⟶}\cdots .}$

Such a sequence is called achain 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 anexact sequence if${\displaystyle \mathrm{im}(f_{i+1})=\ker (f_i)}$. A special case of an exact sequence is a short exact sequence:

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

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

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

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

In the case of modules over acommutative ring, a sequence is exact if and only if it is exact at all themaximal ideals; that is all sequences

${\displaystyle 0\to A_{\mathfrak{m}}\stackrel{f}{\to }{B}_{\mathfrak{m}}\stackrel{g}{\to }{C}_{\mathfrak{m}}\to 0}$

are exact, where the subscript${\displaystyle \mathfrak{m}}$ means thelocalization 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 afiber square (orpullback square), denoted byM ×_BN, if it fits into

${\displaystyle 0\to M{\times }_{B}N\to M\times N\stackrel{\varphi }{\to }B\to 0}$

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

Example: Let${\displaystyle B\subset A}$ be commutative rings, and letI be theannihilator of the quotientB-moduleA/B (which is an ideal ofA). Then canonical maps${\displaystyle A\to A/I,B/I\to A/I}$ form a fiber square with${\displaystyle B=A{\times }_{A/I}B/I.}$

Let${\displaystyle \varphi :M\to M}$ be an endomorphism between finitely generatedR-modules for a commutative ringR. Then

• ${\displaystyle \varphi }$ is killed by its characteristic polynomial relative to the generators ofM; seeNakayama's lemma#Proof.
• If${\displaystyle \varphi }$ is surjective, then it is injective.^[2]

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

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

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

where${\displaystyle D(f)}$ consists of all elementsx inM such that (x,y) belongs tof for somey inN.

Atransgression that arises from a spectral sequence is an example of an additive relation.

• Mapping cone (homological algebra)
• Smith normal form
• Chain complex
• Pairing

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