Module homomorphism

<h2 id="terminology">Terminology</h2>
A module homomorphism is called a module isomorphism if it admits an inverse homomorphism; in particular, it is a <a href="/facts/Bijection/j4gTuTmW">bijection</a>. 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 <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> it is an isomorphism between the underlying abelian groups.
The <a href="/facts/Isomorphism_theorem/SNpPTkrd">isomorphism theorems</a> hold for module homomorphisms.
A module homomorphism from a module M to itself is called an <a href="/facts/Endomorphism/s7NNlSea">endomorphism</a> and an isomorphism from M to itself an <a href="/facts/Automorphism/WnyxNFkQ">automorphism</a>. 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 <a href="/facts/Endomorphism_ring/K1UgpnJK">endomorphism ring</a> of M. The <a href="/facts/Group_of_units/GJSBu8Y7">group of units</a> of this ring is the <a href="/facts/Automorphism_group/65vImKwe">automorphism group</a> of M. 
<a href="/facts/Schur%27s_lemma/JvAT7t09">Schur's lemma</a> says that a homomorphism between <a href="/facts/Simple_module/00roKQrG">simple modules</a> (modules with no non-trivial <a href="/facts/Submodule/jP0LIhFL">submodules</a>) must be either zero or an isomorphism. In particular, the endomorphism ring of a simple module is a <a href="/facts/Division_ring/bCf07uRj">division ring</a>.
In the language of the <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, an injective homomorphism is also called a <a href="/facts/Monomorphism/Q9BrtDTs">monomorphism</a> and a surjective homomorphism an <a href="/facts/Epimorphism/IrzonI01">epimorphism</a>.

<h2 id="examples">Examples</h2>
<ul><li>The <a href="/facts/Zero_map/4NBRtKHc">zero map</a> M → N that maps every element to zero.</li>
<li>A <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformation</a> between <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a>.</li>
<li>
 
 
 
 
 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)}
 
.</li>
<li>For a commutative ring R and <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideals</a> 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}
 
</li></ul>
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 <a href="/facts/Annihilator_(ring_theory)/aTaHR5W7">annihilator</a> of I.
<ul><li>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}
 
.</li></ul>
That is, 
 
 
 
 
 l
 
 r
 
 
 
 
 {\displaystyle l_{r}}
 
 is right R-linear.
<ul><li>For any ring R,
<ul><li>
 
 
 
 
 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 <a href="/facts/Left_regular_representation/dBND2lRj">left regular representation</a> 
 
 
 
 R
 
 
 →
 ∼
 
 
 
 End
 
 R
 
 
 ⁡
 (
 R
 )
 ,
 
 r
 ↦
 
 l
 
 r
 
 
 
 
 {\displaystyle R{\overset {\sim }{\to }}\operatorname {End} _{R}(R),\,r\mapsto l_{r}}
 
.</li>
<li>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.</li>
<li>
 
 
 
 
 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.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> (The module structure on Hom here comes from the right R-action on R; see #Module structures on Hom below.)</li>
<li>
 
 
 
 
 Hom
 
 R
 
 
 ⁡
 (
 M
 ,
 R
 )
 
 
 {\displaystyle \operatorname {Hom} _{R}(M,R)}
 
 is called the <a href="/facts/Dual_module/Q89wXgxc">dual module</a> 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^{*}}
 
.</li></ul></li>
<li>Given a ring homomorphism R → S of commutative rings and an S-module M, an R-linear map θ: S → M is called a <a href="/facts/Derivation_(algebra)/vcSqpxzu">derivation</a> if for any f, g in S, θ(f g) = f θ(g) + θ(f) g.</li>
<li>If S, T are unital <a href="/facts/Associative_algebra/DRfoF6KV">associative algebras</a> over a ring R, then an <a href="/facts/Algebra_homomorphism/b7YL8aPR">algebra homomorphism</a> from S to T is a <a href="/facts/Ring_homomorphism/UptcMpFk">ring homomorphism</a> that is also an R-module homomorphism.</li></ul>
<h2 id="module-structures-on-hom">Module structures on Hom</h2>
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}
 
.

<h2 id="a-matrix-representation">A matrix representation</h2>
The relationship between matrices and linear transformations in <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a> generalizes in a natural way to module homomorphisms between free modules. Precisely, given a right R-module U, there is the <a href="/facts/Canonical_isomorphism/pj8KgkxU">canonical isomorphism</a> 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 <a href="/facts/Matrix_multiplication/lYDB6Ro2">matrix multiplication</a>).
Note the above isomorphism is canonical; no choice is involved. On the other hand, if one is given a module homomorphism between finite-rank <a href="/facts/Free_module/o6rC6yoJ">free modules</a>, 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.

<h2 id="defining">Defining</h2>
In practice, one often defines a module homomorphism by specifying its values on a <a href="/facts/Generating_set_of_a_module/mgvpF9Yo">generating set</a>. More precisely, let M and N be left R-modules. Suppose a <a href="/facts/Subset/SJGERmbA">subset</a> 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 <a href="/facts/Free_presentation/GxxKGm4L">free presentation</a>). 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).

<h2 id="operations">Operations</h2>
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 <a href="/facts/Graph_of_a_function/htYX0BGX">graph</a> Γf of f is the submodule of M ⊕ N given by

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 <a href="/facts/Transpose/8wmsagGS">transpose</a> of f is

f
          
            ∗
          
        
        :
        
          N
          
            ∗
          
        
        →
        
          M
          
            ∗
          
        
        ,
        
        
          f
          
            ∗
          
        
        (
        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.

<h2 id="exact-sequences">Exact sequences</h2>
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 <a href="/facts/Chain_complex/YAlchtCT">chain complex</a> (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., <a href="/facts/De_Rham_complex/4qGKeJq0">de Rham complex</a>.) A chain complex is called an <a href="/facts/Exact_sequence/leUJnwJb">exact sequence</a> 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 <a href="/facts/Cokernel/vYFQRPoh">cokernel</a>, that is the quotient of 
 
 
 
 N
 
 
 {\displaystyle N}
 
 by the image of 
 
 
 
 f
 
 
 {\displaystyle f}
 
.
In the case of modules over a <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a>, a sequence is exact if and only if it is exact at all the <a href="/facts/Maximal_ideal/nf4DnKVS">maximal ideals</a>; 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 <a href="/facts/Localization_of_a_module/rjevzBJf">localization</a> 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 <a href="/facts/Pullback_square/bC0QoX5U">pullback square</a>), denoted by M ×B N, if it fits into

0
        →
        M
        
          ×
          
            B
          
        
        N
        →
        M
        ×
        N
        
          
            →
            ϕ
          
        
        B
        →
        0
      
    
    {\displaystyle 0\to M\times _{B}N\to M\times N{\overset {\phi }{\to }}B\to 0}

where 
 
 
 
 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 <a href="/facts/Annihilator_(ring_theory)/aTaHR5W7">annihilator</a> 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.}

<h2 id="endomorphisms-of-finitely-generated-modules">Endomorphisms of finitely generated modules</h2>
Let 
 
 
 
 ϕ
 :
 M
 →
 M
 
 
 {\displaystyle \phi :M\to M}
 
 be an endomorphism between finitely generated R-modules for a commutative ring R. Then

<ul><li>
 
 
 
 ϕ
 
 
 {\displaystyle \phi }
 
 is killed by its characteristic polynomial relative to the generators of M; see <a href="/facts/Nakayama%27s_lemma/Nwhck7ru">Nakayama's lemma#Proof</a>.</li>
<li>If 
 
 
 
 ϕ
 
 
 {\displaystyle \phi }
 
 is surjective, then it is injective.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a></li></ul>
See also: <a href="/facts/Herbrand_quotient/LH8wHBnt">Herbrand quotient</a> (which can be defined for any endomorphism with some finiteness conditions.)

<h2 id="variant-additive-relations">Variant: additive relations</h2>
See also: <a href="/facts/Binary_relation/r9BwGgaK">binary relation</a>
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.}
 
<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> In other words, it is a "<a href="/facts/Many-valued_function/WH5ua6BL">many-valued</a>" 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 <a href="/facts/Spectral_sequence/9S1v2dRh">transgression</a> that arises from a spectral sequence is an example of an additive relation.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Mapping_cone_(homological_algebra)/fw2WKVfY">Mapping cone (homological algebra)</a></li>
<li><a href="/facts/Smith_normal_form/pchMkW7Q">Smith normal form</a></li>
<li><a href="/facts/Chain_complex/YAlchtCT">Chain complex</a></li>
<li><a href="/facts/Pairing/G0qgKRD8">Pairing</a></li></ul>
<h2 id="notes">Notes</h2>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Bourbaki, Nicolas (1998), "Chapter II, §1.14, remark 2", Algebra I, Chapters 1–3, Elements of Mathematics, Springer-Verlag, ISBN 3-540-64243-9, MR 1727844 <a href="3-540-64243-9" target="_blank">3-540-64243-9</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Matsumura, Hideyuki (1989), "Theorem 2.4", Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, vol. 8 (2nd ed.), Cambridge University Press, ISBN 0-521-36764-6, MR 1011461 <a href="0-521-36764-6" target="_blank">0-521-36764-6</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Mac Lane, Saunders (1995), Homology, Classics in Mathematics, Springer-Verlag, p. 52, ISBN 3-540-58662-8, MR 1344215 <a href="3-540-58662-8" target="_blank">3-540-58662-8</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Module homomorphism open-in-new

Module homomorphism