`:top
In `F33f`_`[algebra`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Abstract_algebra]`_`f, a `!module homomorphism`! is a `F33f`_`[function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_(mathematics)]`_`f between `F33f`_`[modules`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Module_(mathematics)]`_`f that preserves the module structures. Explicitly, if `*M`* and `*N`* are left modules over a `F33f`_`[ring`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ring_(mathematics)]`_`f `*R`*, then a function f : M → → N {\\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`*,
f ( x + y ) = f ( x ) + f ( y ) , {\\displaystyle f(x+y)=f(x)+f(y),}
f ( r x ) = r f ( x ) . {\\displaystyle f(rx)=rf(x).}
In other words, `*f`* is a `F33f`_`[group homomorphism`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Group_homomorphism]`_`f (for the underlying additive groups) that commutes with `F33f`_`[scalar multiplication`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Scalar_multiplication]`_`f. If `*M`*, `*N`* are right `*R`*-modules, then the second condition is replaced with
f ( x r ) = f ( x ) r . {\\displaystyle f(xr)=f(x)r.}
The `F33f`_`[preimage`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Preimage]`_`f of the zero element under `*f`* is called the `F33f`_`[kernel`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Kernel_(algebra)]`_`f of `*f`*. The `F33f`_`[set`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Set_(mathematics)]`_`f of all module homomorphisms from `*M`* to `*N`* is denoted by Hom R ( M , N ) {\\displaystyle \\operatorname {Hom} _{R}(M,N)} . It is an `F33f`_`[abelian group`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Abelian_group]`_`f (under pointwise addition) but is not necessarily a module unless `*R`* is `F33f`_`[commutative`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Commutative_ring]`_`f.
The `F33f`_`[composition`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_composition]`_`f 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 `F33f`_`[category of modules`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Category_of_modules]`_`f.
>>Contents
• `F0af`_`[Terminology`#terminology]`_`f
• `F0af`_`[Examples`#examples]`_`f
• `F0af`_`[Module structures on Hom`#module-structures-on-hom]`_`f
• `F0af`_`[A matrix representation`#a-matrix-representation]`_`f
• `F0af`_`[Defining`#defining]`_`f
• `F0af`_`[Operations`#operations]`_`f
• `F0af`_`[Exact sequences`#exact-sequences]`_`f
• `F0af`_`[Endomorphisms of finitely generated modules`#endomorphisms-of-finitely-generated-modules]`_`f
• `F0af`_`[Variant: additive relations`#variant-additive-relations]`_`f
• `F0af`_`[See also`#see-also]`_`f
• `F0af`_`[Notes`#notes]`_`f
-─
>>Terminology
A module homomorphism is called a `*module isomorphism`* if it admits an inverse homomorphism; in particular, it is a `F33f`_`[bijection`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bijection]`_`f. 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 `F33f`_`[if and only if`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=If_and_only_if]`_`f it is an isomorphism between the underlying abelian groups.
The `F33f`_`[isomorphism theorems`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Isomorphism_theorem]`_`f hold for module homomorphisms.
A module homomorphism from a module `*M`* to itself is called an `F33f`_`[endomorphism`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Endomorphism]`_`f and an isomorphism from `*M`* to itself an `F33f`_`[automorphism`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Automorphism]`_`f. One writes End R ( M ) = Hom R ( M , M ) {\\displaystyle \\operatorname {End} _{R}(M)=\\operatorname {Hom} _{R}(M,M)} for the set of all endomorphisms of a module `*M`*. It is not only an abelian group but is also a ring with multiplication given by function composition, called the `F33f`_`[endomorphism ring`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Endomorphism_ring]`_`f of `*M`*. The `F33f`_`[group of units`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Group_of_units]`_`f of this ring is the `F33f`_`[automorphism group`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Automorphism_group]`_`f of `*M`*.
`F33f`_`[Schur's lemma`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Schur's_lemma]`_`f says that a homomorphism between `F33f`_`[simple modules`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Simple_module]`_`f (modules with no non-trivial `F33f`_`[submodules`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Submodule]`_`f) must be either zero or an isomorphism. In particular, the endomorphism ring of a simple module is a `F33f`_`[division ring`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Division_ring]`_`f.
In the language of the `F33f`_`[category theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Category_theory]`_`f, an injective homomorphism is also called a `F33f`_`[monomorphism`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Monomorphism]`_`f and a surjective homomorphism an `F33f`_`[epimorphism`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Epimorphism]`_`f.
>>Examples
• The `F33f`_`[zero map`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Zero_map]`_`f `*M`* → `*N`* that maps every element to zero.
• A `F33f`_`[linear transformation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Linear_transformation]`_`f between `F33f`_`[vector spaces`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Vector_space]`_`f.
• Hom Z ( Z / n , Z / m ) = Z / gcd ( n , m ) {\\displaystyle \\operatorname {Hom} _{\\mathbb {Z} }(\\mathbb {Z} /n,\\mathbb {Z} /m)=\\mathbb {Z} /\\operatorname {gcd} (n,m)} .
• For a commutative ring `*R`* and `F33f`_`[ideals`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ideal_(ring_theory)]`_`f `*I`*, `*J`*, there is the canonical identification Hom R ( R / I , R / J ) = { r ∈ ∈ R | r I ⊂ ⊂ J } / J {\\displaystyle \\operatorname {Hom} _{R}(R/I,R/J)=\\{r\\in R|rI\\subset J\\}/J}
given by f ↦ ↦ f ( 1 ) {\\displaystyle f\\mapsto f(1)} . In particular, Hom R ( R / I , R ) {\\displaystyle \\operatorname {Hom} _{R}(R/I,R)} is the `F33f`_`[annihilator`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Annihilator_(ring_theory)]`_`f of `*I`*.
• Given a ring `*R`* and an element `*r`*, let l r : R → → R {\\displaystyle l_{r}:R\\to 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 {\\displaystyle l_{r}(st)=rst=l_{r}(s)t} .
That is, l r {\\displaystyle l_{r}} is `*right`* `*R`*-linear.
• For any ring `*R`*,
• End R ( R ) = 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 `F33f`_`[left regular representation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Left_regular_representation]`_`f R → → ∼ ∼ End R ( R ) , r ↦ ↦ l r {\\displaystyle R{\\overset {\\sim }{\\to }}\\operatorname {End} _{R}(R),\\,r\\mapsto l_{r}} .
• Similarly, End R ( R ) = R o p {\\displaystyle \\operatorname {End} _{R}(R)=R^{op}} as rings when `*R`* is viewed as a left module over itself. Textbooks or other references usually specify which convention is used.
• Hom R ( R , M ) = M {\\displaystyle \\operatorname {Hom} _{R}(R,M)=M} through f ↦ ↦ f ( 1 ) {\\displaystyle f\\mapsto f(1)} for any left module `*M`*.`:cite-ref-bourbaki-1-0[`F5bf`_`[1`#cite-note-bourbaki-1]`_`f] (The module structure on Hom here comes from the right `*R`*-action on `*R`*; see `F33f`_`[#Module structures on Hom`#module-structures-on-hom]`_`f below.)
• Hom R ( M , R ) {\\displaystyle \\operatorname {Hom} _{R}(M,R)} is called the `F33f`_`[dual module`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dual_module]`_`f 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 ∗ ∗ {\\displaystyle 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 `F33f`_`[derivation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Derivation_(algebra)]`_`f if for any `*f`*, `*g`* in `*S`*, θ(`*f g`*) = `*f`* θ(`*g`*) + θ(`*f`*) `*g`*.
• If `*S`*, `*T`* are unital `F33f`_`[associative algebras`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Associative_algebra]`_`f over a ring `*R`*, then an `F33f`_`[algebra homomorphism`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebra_homomorphism]`_`f from `*S`* to `*T`* is a `F33f`_`[ring homomorphism`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ring_homomorphism]`_`f 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 ) {\\displaystyle \\operatorname {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 ) . {\\displaystyle (s\\cdot f)(x)=f(xs).}
It is well-defined (i.e., s ⋅ ⋅ f {\\displaystyle s\\cdot f} is `*R`*-linear) since
( s ⋅ ⋅ f ) ( r x ) = f ( r x s ) = r f ( x s ) = r ( s ⋅ ⋅ f ) ( x ) , {\\displaystyle (s\\cdot f)(rx)=f(rxs)=rf(xs)=r(s\\cdot f)(x),}
and s ⋅ ⋅ f {\\displaystyle s\\cdot f} is a ring action since
( s t ⋅ ⋅ f ) ( x ) = f ( x s t ) = ( t ⋅ ⋅ f ) ( x s ) = s ⋅ ⋅ ( t ⋅ ⋅ f ) ( x ) {\\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 Hom R ( M , N ) {\\displaystyle \\operatorname {Hom} _{R}(M,N)} is a right `*S`*-module by ( f ⋅ ⋅ s ) ( x ) = f ( x ) s {\\displaystyle (f\\cdot s)(x)=f(x)s} .
>>A matrix representation
The relationship between matrices and linear transformations in `F33f`_`[linear algebra`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Linear_algebra]`_`f generalizes in a natural way to module homomorphisms between free modules. Precisely, given a right `*R`*-module `*U`*, there is the `F33f`_`[canonical isomorphism`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Canonical_isomorphism]`_`f of the abelian groups
Hom R ( U ⊕ ⊕ n , U ⊕ ⊕ m ) → → ∼ ∼ f ↦ ↦ [ f i j ] M m , n ( End R ( U ) ) {\\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 U ⊕ ⊕ n {\\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 End R ( R ) ≃ ≃ R {\\displaystyle \\operatorname {End} _{R}(R)\\simeq R} , one has
End R ( R n ) ≃ ≃ M n ( R ) {\\displaystyle \\operatorname {End} _{R}(R^{n})\\simeq M_{n}(R)} ,
which turns out to be a ring isomorphism (as a composition corresponds to a `F33f`_`[matrix multiplication`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Matrix_multiplication]`_`f).
Note the above isomorphism is canonical; no choice is involved. On the other hand, if one is given a module homomorphism between finite-rank `F33f`_`[free modules`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Free_module]`_`f, then a choice of an ordered basis corresponds to a choice of an isomorphism F ≃ ≃ R n {\\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 `F33f`_`[generating set`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Generating_set_of_a_module]`_`f. More precisely, let `*M`* and `*N`* be left `*R`*-modules. Suppose a `F33f`_`[subset`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Subset]`_`f `*S`* generates `*M`*; i.e., there is a surjection F → → M {\\displaystyle F\\to M} with a free module `*F`* with a basis indexed by `*S`* and kernel `*K`* (i.e., one has a `F33f`_`[free presentation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Free_presentation]`_`f). Then to give a module homomorphism M → → N {\\displaystyle M\\to N} is to give a module homomorphism F → → N {\\displaystyle F\\to N} that kills `*K`* (i.e., maps `*K`* to zero).
>>Operations
If f : M → → N {\\displaystyle f:M\\to N} and g : M ′ → → N ′ {\\displaystyle g:M'\\to N'} are module homomorphisms, then their direct sum is
f ⊕ ⊕ g : M ⊕ ⊕ M ′ → → N ⊕ ⊕ N ′ , ( x , y ) ↦ ↦ ( f ( x ) , g ( y ) ) {\\displaystyle f\\oplus g:M\\oplus M'\\to N\\oplus N',\\,(x,y)\\mapsto (f(x),g(y))}
and their tensor product is
f ⊗ ⊗ g : M ⊗ ⊗ M ′ → → N ⊗ ⊗ N ′ , x ⊗ ⊗ y ↦ ↦ f ( x ) ⊗ ⊗ g ( y ) . {\\displaystyle f\\otimes g:M\\otimes M'\\to N\\otimes N',\\,x\\otimes y\\mapsto f(x)\\otimes g(y).}
Let f : M → → N {\\displaystyle f:M\\to N} be a module homomorphism between left modules. The `F33f`_`[graph`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Graph_of_a_function]`_`f Γ`*f`* of `*f`* is the submodule of `*M`* ⊕ `*N`* given by
Γ Γ f = { ( x , f ( x ) ) | x ∈ ∈ M } {\\displaystyle \\Gamma _{f}=\\{(x,f(x))|x\\in M\\}} ,
which is the image of the module homomorphism `*M`* → `*M`* ⊕ `*N`*, `*x`* → (`*x`*, `*f`*(`*x`*)), called the `!graph morphism`!.
The `F33f`_`[transpose`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Transpose]`_`f of `*f`* is
f ∗ ∗ : N ∗ ∗ → → M ∗ ∗ , f ∗ ∗ ( α α ) = α α ∘ ∘ f . {\\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
⋯ ⋯ ⟶ ⟶ f 3 M 2 ⟶ ⟶ f 2 M 1 ⟶ ⟶ f 1 M 0 ⟶ ⟶ f 0 M − − 1 ⟶ ⟶ f − − 1 ⋯ ⋯ . {\\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 `F33f`_`[chain complex`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Chain_complex]`_`f (or often just complex) if each composition is zero; i.e., f i ∘ ∘ f i + 1 = 0 {\\displaystyle f_{i}\\circ f_{i+1}=0} or equivalently the image of f i + 1 {\\displaystyle f_{i+1}} is contained in the kernel of f i {\\displaystyle f_{i}} . (If the numbers increase instead of decrease, then it is called a cochain complex; e.g., `F33f`_`[de Rham complex`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=De_Rham_complex]`_`f.) A chain complex is called an `F33f`_`[exact sequence`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Exact_sequence]`_`f if im ( f i + 1 ) = ker ( f i ) {\\displaystyle \\operatorname {im} (f_{i+1})=\\operatorname {ker} (f_{i})} . A special case of an exact sequence is a short exact sequence:
0 → → A → → f B → → g C → → 0 {\\displaystyle 0\\to A{\\overset {f}{\\to }}B{\\overset {g}{\\to }}C\\to 0}
where f {\\displaystyle f} is injective, the kernel of g {\\displaystyle g} is the image of f {\\displaystyle f} and g {\\displaystyle g} is surjective.
Any module homomorphism f : M → → N {\\displaystyle f:M\\to N} defines an exact sequence
0 → → K → → M → → f N → → C → → 0 , {\\displaystyle 0\\to K\\to M{\\overset {f}{\\to }}N\\to C\\to 0,}
where K {\\displaystyle K} is the kernel of f {\\displaystyle f} , and C {\\displaystyle C} is the `F33f`_`[cokernel`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cokernel]`_`f, that is the quotient of N {\\displaystyle N} by the image of f {\\displaystyle f} .
In the case of modules over a `F33f`_`[commutative ring`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Commutative_ring]`_`f, a sequence is exact if and only if it is exact at all the `F33f`_`[maximal ideals`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Maximal_ideal]`_`f; that is all sequences
0 → → A m → → f B m → → g C m → → 0 {\\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 m {\\displaystyle {\\mathfrak {m}}} means the `F33f`_`[localization`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Localization_of_a_module]`_`f at a maximal ideal m {\\displaystyle {\\mathfrak {m}}} .
If f : M → → B , g : N → → B {\\displaystyle f:M\\to B,g:N\\to B} are module homomorphisms, then they are said to form a `!fiber square`! (or `!`F33f`_`[pullback square`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pullback_square]`_`f`!), denoted by `*M`* ×`*B`* `*N`*, if it fits into
0 → → M × × B N → → M × × N → → ϕ ϕ B → → 0 {\\displaystyle 0\\to M\\times _{B}N\\to M\\times N{\\overset {\\phi }{\\to }}B\\to 0}
where ϕ ϕ ( x , y ) = f ( x ) − − g ( x ) {\\displaystyle \\phi (x,y)=f(x)-g(x)} .
Example: Let B ⊂ ⊂ A {\\displaystyle B\\subset A} be commutative rings, and let `*I`* be the `F33f`_`[annihilator`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Annihilator_(ring_theory)]`_`f of the quotient `*B`*-module `*A`*/`*B`* (which is an ideal of `*A`*). Then canonical maps A → → A / I , B / I → → A / I {\\displaystyle A\\to A/I,B/I\\to A/I} form a fiber square with B = A × × A / I B / I . {\\displaystyle B=A\\times _{A/I}B/I.}
>>Endomorphisms of finitely generated modules
Let ϕ ϕ : M → → M {\\displaystyle \\phi :M\\to M} be an endomorphism between finitely generated `*R`*-modules for a commutative ring `*R`*. Then
• ϕ ϕ {\\displaystyle \\phi } is killed by its characteristic polynomial relative to the generators of `*M`*; see `F33f`_`[Nakayama's lemma#Proof`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Nakayama's_lemma]`_`f.
• If ϕ ϕ {\\displaystyle \\phi } is surjective, then it is injective.`:cite-ref-matsumura-2-0[`F5bf`_`[2`#cite-note-matsumura-2]`_`f]
See also: `F33f`_`[Herbrand quotient`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Herbrand_quotient]`_`f (which can be defined for any endomorphism with some finiteness conditions.)
>>Variant: additive relations
An `!additive relation`! M → → N {\\displaystyle M\\to N} from a module `*M`* to a module `*N`* is a submodule of M ⊕ ⊕ N . {\\displaystyle M\\oplus N.} `:cite-ref-maclane-3-0[`F5bf`_`[3`#cite-note-maclane-3]`_`f] In other words, it is a "`F33f`_`[many-valued`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Many-valued_function]`_`f" homomorphism defined on some submodule of `*M`*. The inverse f − − 1 {\\displaystyle f^{-1}} of `*f`* is the submodule { ( y , x ) | ( x , y ) ∈ ∈ f } {\\displaystyle \\{(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 ) → → N / { y | ( 0 , y ) ∈ ∈ f } {\\displaystyle D(f)\\to N/\\{y|(0,y)\\in f\\}}
where D ( f ) {\\displaystyle D(f)} consists of all elements `*x`* in `*M`* such that (`*x`*, `*y`*) belongs to `*f`* for some `*y`* in `*N`*.
A `F33f`_`[transgression`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Spectral_sequence]`_`f that arises from a spectral sequence is an example of an additive relation.
>>See also
• `F33f`_`[Mapping cone (homological algebra)`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mapping_cone_(homological_algebra)]`_`f
• `F33f`_`[Smith normal form`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Smith_normal_form]`_`f
• `F33f`_`[Chain complex`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Chain_complex]`_`f
• `F33f`_`[Pairing`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pairing]`_`f
>>Notes
`:cite-note-bourbaki-1`!1.`! `F0af`_`[↑`#cite-ref-bourbaki-1-0]`_`f `:citerefbourbaki1998`a`F33f`_`[Bourbaki, Nicolas`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Nicolas_Bourbaki]`_`f (1998), "Chapter II, §1.14, remark 2", `*Algebra I, Chapters 1–3`*, Elements of Mathematics, Springer-Verlag, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 3-540-64243-9, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 1727844
`:cite-note-matsumura-2`!2.`! `F0af`_`[↑`#cite-ref-matsumura-2-0]`_`f `:citerefmatsumura1989`aMatsumura, Hideyuki (1989), "Theorem 2.4", `*Commutative Ring Theory`*, Cambridge Studies in Advanced Mathematics, vol. 8 (2nd ed.), Cambridge University Press, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-521-36764-6, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 1011461
`:cite-note-maclane-3`!3.`! `F0af`_`[↑`#cite-ref-maclane-3-0]`_`f `:citerefmac-lane1995`a`F33f`_`[Mac Lane, Saunders`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Saunders_Mac_Lane]`_`f (1995), `*Homology`*, Classics in Mathematics, Springer-Verlag, p. 52, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 3-540-58662-8, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 1344215
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