Glossary of module theory

algebraically compact
<a href="/facts/Algebraically_compact_module/wL0ugvd0">algebraically compact module</a> (also called <a href="/facts/Pure_injective_module/wL0ugvd0">pure injective module</a>) is a module in which all systems of equations can be decided by finitary means. Alternatively, those modules which leave pure-exact sequence exact after applying Hom.

annihilator
1.  The <a href="/facts/Annihilator_(ring_theory)/aTaHR5W7">annihilator</a> of a left 
  
    
      
        R
      
    
    {\displaystyle R}
  
-module 
  
    
      
        M
      
    
    {\displaystyle M}
  
 is the set 
  
    
      
        
          
            Ann
          
        
        (
        M
        )
        :=
        {
        r
        ∈
        R
         
        
          |
        
         
        r
        m
        =
        0
        
        ∀
        m
        ∈
        M
        }
      
    
    {\displaystyle {\textrm {Ann}}(M):=\{r\in R~|~rm=0\,\forall m\in M\}}
  
  . It is a (left) <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a> of 
  
    
      
        R
      
    
    {\displaystyle R}
  
.
2.  The annihilator of an element 
  
    
      
        m
        ∈
        M
      
    
    {\displaystyle m\in M}
  
 is the set 
  
    
      
        
          
            Ann
          
        
        (
        m
        )
        :=
        {
        r
        ∈
        R
         
        
          |
        
         
        r
        m
        =
        0
        }
      
    
    {\displaystyle {\textrm {Ann}}(m):=\{r\in R~|~rm=0\}}
  
.

Artinian
An <a href="/facts/Artinian_module/jbg1NSk5">Artinian module</a> is a module in which every decreasing chain of submodules becomes stationary after finitely many steps.

associated prime
1.  <a href="/facts/Associated_prime/PtXW8lal">associated prime</a>

automorphism
An <a href="/facts/Automorphism/WnyxNFkQ">automorphism</a> is an <a href="/facts/Endomorphism/s7NNlSea">endomorphism</a> that is also an isomorphism.

Azumaya
<a href="/facts/Azumaya%2527s_theorem/9ujxGNUW">Azumaya's theorem</a> says that two decompositions into modules with local endomorphism rings are equivalent.

balanced
<a href="/facts/Balanced_module/RuKIusf1">balanced module</a>

basis
A basis of a module 
  
    
      
        M
      
    
    {\displaystyle M}
  
 is a set of elements in 
  
    
      
        M
      
    
    {\displaystyle M}
  
 such that every element in the module can be expressed as a finite sum of elements in the basis in a unique way.

Beauville–Laszlo
<a href="/facts/Beauville%25E2%2580%2593Laszlo_theorem/JXxTzREV">Beauville–Laszlo theorem</a>

big
"big" usually means "not-necessarily finitely generated".

bimodule
<a href="/facts/Bimodule/NskmNGQ1">bimodule</a>

canonical module
<a href="/facts/Canonical_module/W6XrJr5R">canonical module</a> (the term "canonical" comes from <a href="/facts/Canonical_divisor/nJKN8mpG">canonical divisor</a>)

category
The <a href="/facts/Category_of_modules/AnnF9JKf">category of modules</a> over a ring is the category where the objects are all the (say) left modules over the given ring and the morphisms module homomorphisms.

character
<a href="/facts/Character_module/Sa37BxRD">character module</a>

chain complex
<a href="/facts/Chain_complex/YAlchtCT">chain complex</a> (frequently just complex)

closed submodule
A module is called a closed submodule if it does not contain any <a href="/facts/Essential_extension/S50wKkfs">essential extension</a>.

Cohen–Macaulay
<a href="/facts/Cohen%25E2%2580%2593Macaulay_module/4G8wymtC">Cohen–Macaulay module</a>

coherent
A <a href="/facts/Coherent_module/6Njcpgzr">coherent module</a> is a finitely generated module whose finitely generated submodules are <a href="/facts/Finitely_presented_module/6Njcpgzr">finitely presented</a>.

cokernel
The <a href="/facts/Cokernel/vYFQRPoh">cokernel</a> of a module homomorphism is the codomain quotiented by the image.

compact
A compact module

completely reducible
Synonymous to "<a href="/facts/Semisimple_module/9KASYk35">semisimple module</a>".

completion
<a href="/facts/Completion_(ring_theory)/Q0vjA9JI">completion</a> of a module

composition
<a href="/facts/Composition_series/3C7CHH3k">Jordan Hölder composition series</a>

continuous
<a href="/facts/Continuous_module/sTCpn4tD">continuous module</a>

countably generated
A <a href="/facts/Countably_generated_module/b9c4FnWJ">countably generated module</a> is a module that admits a generating set whose cardinality is at most countable.

cyclic
A module is called a <a href="/facts/Cyclic_module/w9tMplLC">cyclic module</a> if it is generated by one element.

D
A <a href="/facts/D-module/l1y6evYe">D-module</a> is a module over a ring of differential operators.

decomposition
A <a href="/facts/Decomposition_of_a_module/9ujxGNUW">decomposition of a module</a> is a way to express a module as a direct sum of submodules.

dense
dense submodule

determinant
The determinant of a finite free module over a commutative ring is the <i>r</i>-th exterior power of the module when <i>r</i> is the rank of the module.

differential
A <a href="/facts/Differential_graded_module/5L5fG2bv">differential graded module</a> or dg-module is a graded module with a differential.

direct sum
A <a href="/facts/Direct_sum_of_modules/Y6PBJnjg">direct sum of modules</a> is a module that is the direct sum of the underlying abelian group together with component-wise scalar multiplication.

dual module
The dual module of a module <i>M</i> over a commutative ring <i>R</i> is the module 
  
    
      
        
          Hom
          
            R
          
        
        ⁡
        (
        M
        ,
        R
        )
      
    
    {\displaystyle \operatorname {Hom} _{R}(M,R)}
  
.

dualizing
<a href="/facts/Dualizing_module/W6XrJr5R">dualizing module</a>

Drinfeld
A <a href="/facts/Drinfeld_module/WMJkFbYB">Drinfeld module</a> is a module over a ring of functions on algebraic curve with coefficients from a finite field.

Eilenberg–Mazur
<a href="/facts/Eilenberg%25E2%2580%2593Mazur_swindle/AdBQTVzz">Eilenberg–Mazur swindle</a>

elementary
<a href="/facts/Elementary_divisor/GPk7IPuL">elementary divisor</a>

endomorphism
1.  An <a href="/facts/Endomorphism/s7NNlSea">endomorphism</a> is a module homomorphism from a module to itself.
2.  The <a href="/facts/Endomorphism_ring/K1UgpnJK">endomorphism ring</a> is the set of all module homomorphisms with addition as addition of functions and multiplication composition of functions.

enough
<a href="/facts/Enough_injectives/JGABzfyw">enough injectives</a>
<a href="/facts/Enough_projectives/sKWPFB6u">enough projectives</a>

essential
Given a module <i>M</i>, an <a href="/facts/Essential_submodule/S50wKkfs">essential submodule</a> <i>N</i> of <i>M</i> is a submodule that every nonzero submodule of <i>M</i> intersects non-trivially.

exact
<a href="/facts/Exact_sequence/leUJnwJb">exact sequence</a>

Ext functor
<a href="/facts/Ext_functor/ktggDk7I">Ext functor</a>

extension
<a href="/facts/Extension_of_scalars/RYGsTsYk">Extension of scalars</a> uses a ring homomorphism from <i>R</i> to <i>S</i> to convert <i>R</i>-modules to <i>S</i>-modules.

faithful
A <a href="/facts/Faithful_module/aTaHR5W7">faithful module</a> 
  
    
      
        M
      
    
    {\displaystyle M}
  
 is one where the action of each nonzero 
  
    
      
        r
        ∈
        R
      
    
    {\displaystyle r\in R}
  
 on 
  
    
      
        M
      
    
    {\displaystyle M}
  
 is nontrivial (i.e. 
  
    
      
        r
        x
        ≠
        0
      
    
    {\displaystyle rx\neq 0}
  
 for some 
  
    
      
        x
      
    
    {\displaystyle x}
  
 in 
  
    
      
        M
      
    
    {\displaystyle M}
  
). Equivalently, 
  
    
      
        
          
            Ann
          
        
        (
        M
        )
      
    
    {\displaystyle {\textrm {Ann}}(M)}
  
 is the zero ideal.

finite
The term "<a href="/facts/Finite_module/6Njcpgzr">finite module</a>" is another name for a <a href="/facts/Finitely_generated_module/6Njcpgzr">finitely generated module</a>.

finite length
A module of finite <a href="/facts/Length_of_a_module/UpWodT4k">length</a> is a module that admits a (finite) composition series.

finite presentation
1.  A <a href="/facts/Finite_free_presentation/GxxKGm4L">finite free presentation</a> of a module <i>M</i> is an exact sequence 
  
    
      
        
          F
          
            1
          
        
        →
        
          F
          
            0
          
        
        →
        M
      
    
    {\displaystyle F_{1}\to F_{0}\to M}
  
 where 
  
    
      
        
          F
          
            i
          
        
      
    
    {\displaystyle F_{i}}
  
 are finitely generated free modules.
2.  A <a href="/facts/Finitely_presented_module/6Njcpgzr">finitely presented module</a> is a module that admits a <a href="/facts/Finite_free_presentation/GxxKGm4L">finite free presentation</a>.

finitely generated
A module 
  
    
      
        M
      
    
    {\displaystyle M}
  
 is <a href="/facts/Finitely_generated_module/6Njcpgzr">finitely generated</a> if there exist finitely many elements 
  
    
      
        
          x
          
            1
          
        
        ,
        .
        .
        .
        ,
        
          x
          
            n
          
        
      
    
    {\displaystyle x_{1},...,x_{n}}
  
 in 
  
    
      
        M
      
    
    {\displaystyle M}
  
 such that every element of 
  
    
      
        M
      
    
    {\displaystyle M}
  
 is a finite linear combination of those elements with coefficients from the scalar ring 
  
    
      
        R
      
    
    {\displaystyle R}
  
.

fitting
1.  <a href="/facts/Fitting_ideal/TR5K1NIc">fitting ideal</a>
2.  <a href="/facts/Fitting%2527s_lemma/SaV3luE8">Fitting's lemma</a>

five
<a href="/facts/Five_lemma/VblzvyzS">Five lemma</a>

flat
A 
  
    
      
        R
      
    
    {\displaystyle R}
  
-module 
  
    
      
        F
      
    
    {\displaystyle F}
  
 is called a <a href="/facts/Flat_module/1WFrnJP3">flat module</a> if the <a href="/facts/Tensor_product_of_modules/K3oaFIS4">tensor product</a> functor 
  
    
      
        −
        
          ⊗
          
            R
          
        
        F
      
    
    {\displaystyle -\otimes _{R}F}
  
 is <a href="/facts/Exact_functor/4tsQzwBp">exact</a>.In particular, every projective module is flat.

free
A <a href="/facts/Free_module/o6rC6yoJ">free module</a> is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the scalar ring 
  
    
      
        R
      
    
    {\displaystyle R}
  
.

Frobenius reciprocity
<a href="/facts/Frobenius_reciprocity/qoLTxRCX">Frobenius reciprocity</a>.

Galois
A <a href="/facts/Galois_module/fYDgqs8d">Galois module</a> is a module over the group ring of a Galois group.

generating set
A subset of a module is called a <a href="/facts/Generating_set_of_a_module/mgvpF9Yo">generating set</a> of the module if the submodule generated by the set (i.e., the smallest subset containing the set) is the entire module itself.

global
<a href="/facts/Global_dimension/73RGdPyI">global dimension</a>

graded
A module 
  
    
      
        M
      
    
    {\displaystyle M}
  
 over a graded ring 
  
    
      
        A
        =
        
          ⨁
          
            n
            ∈
            
              N
            
          
        
        
          A
          
            n
          
        
      
    
    {\displaystyle A=\bigoplus _{n\in \mathbb {N} }A_{n}}
  
 is a <a href="/facts/Graded_module/AIUjkSaP">graded module</a> if 
  
    
      
        M
      
    
    {\displaystyle M}
  
 can be expressed as a direct sum 
  
    
      
        
          ⨁
          
            i
            ∈
            
              N
            
          
        
        
          M
          
            i
          
        
      
    
    {\displaystyle \bigoplus _{i\in \mathbb {N} }M_{i}}
  
 and 
  
    
      
        
          A
          
            i
          
        
        
          M
          
            j
          
        
        ⊆
        
          M
          
            i
            +
            j
          
        
      
    
    {\displaystyle A_{i}M_{j}\subseteq M_{i+j}}
  
.

Herbrand quotient
A <a href="/facts/Herbrand_quotient/LH8wHBnt">Herbrand quotient</a> of a module homomorphism is another term for index.

Hilbert
1.  <a href="/facts/Hilbert%2527s_syzygy_theorem/HtYbyvz2">Hilbert's syzygy theorem</a>
2.  The <a href="/facts/Hilbert%25E2%2580%2593Poincar%25C3%25A9_series/cVJBU8yR">Hilbert–Poincaré series</a> of a graded module.
3.  The <a href="/facts/Hilbert%25E2%2580%2593Serre_theorem/cVJBU8yR">Hilbert–Serre theorem</a> tells when a Hilbert–Poincaré series is a rational function.

homological dimension
<a href="/facts/Homological_dimension/aID3MIXc">homological dimension</a>

homomorphism
For two left 
  
    
      
        R
      
    
    {\displaystyle R}
  
-modules 
  
    
      
        
          M
          
            1
          
        
        ,
        
          M
          
            2
          
        
      
    
    {\displaystyle M_{1},M_{2}}
  
, a group homomorphism 
  
    
      
        ϕ
        :
        
          M
          
            1
          
        
        →
        
          M
          
            2
          
        
      
    
    {\displaystyle \phi :M_{1}\to M_{2}}
  
 is called <a href="/facts/Module_homomorphism/SCSzvgxx">homomorphism of 
  
    
      
        R
      
    
    {\displaystyle R}
  
-modules</a> if 
  
    
      
        r
        ϕ
        (
        m
        )
        =
        ϕ
        (
        r
        m
        )
        
        ∀
        r
        ∈
        R
        ,
        m
        ∈
        
          M
          
            1
          
        
      
    
    {\displaystyle r\phi (m)=\phi (rm)\,\forall r\in R,m\in M_{1}}
  
 .

Hom
<a href="/facts/Hom_functor/H76BLKCD">Hom functor</a>

idempotent
An <a href="/facts/Idempotent_endomorphism/RE6AkdQd">idempotent</a> is an endomorphism whose square is itself.

indecomposable
An <a href="/facts/Indecomposable_module/N9B1gpzI">indecomposable module</a> is a non-zero module that cannot be written as a direct sum of two non-zero submodules. Every simple module is indecomposable (but not conversely).

index
The index of an endomorphism 
  
    
      
        f
        :
        M
        →
        M
      
    
    {\displaystyle f:M\to M}
  
 is the difference 
  
    
      
        length
        ⁡
        (
        coker
        ⁡
        (
        f
        )
        )
        −
        length
        ⁡
        (
        ker
        ⁡
        (
        f
        )
        )
      
    
    {\displaystyle \operatorname {length} (\operatorname {coker} (f))-\operatorname {length} (\operatorname {ker} (f))}
  
, when the cokernel and kernel of 
  
    
      
        f
      
    
    {\displaystyle f}
  
 have finite length.

injective
1.  A 
  
    
      
        R
      
    
    {\displaystyle R}
  
-module 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
 is called an <a href="/facts/Injective_module/1sAYFnyJ">injective module</a> if given a 
  
    
      
        R
      
    
    {\displaystyle R}
  
-module homomorphism 
  
    
      
        g
        :
        X
        →
        Q
      
    
    {\displaystyle g:X\to Q}
  
, and an <a href="/facts/Injective/5ES6tqf5">injective</a> 
  
    
      
        R
      
    
    {\displaystyle R}
  
-module homomorphism 
  
    
      
        f
        :
        X
        →
        Y
      
    
    {\displaystyle f:X\to Y}
  
, there exists a

R
      
    
    {\displaystyle R}
  
-module homomorphism 
  
    
      
        h
        :
        Y
        →
        Q
      
    
    {\displaystyle h:Y\to Q}
  
 such that 
  
    
      
        f
        ∘
        h
        =
        g
      
    
    {\displaystyle f\circ h=g}
  
.

The following conditions are equivalent:
<ul><li>The contravariant functor 
  
    
      
        
          
            
              Hom
            
          
          
            R
          
        
        (
        −
        ,
        I
        )
      
    
    {\displaystyle {\textrm {Hom}}_{R}(-,I)}
  
 is <a href="/facts/Exact_functor/4tsQzwBp">exact</a>.</li>
<li>
  
    
      
        I
      
    
    {\displaystyle I}
  
 is a injective module.</li>
<li>Every short exact sequence 
  
    
      
        0
        →
        I
        →
        L
        →
        
          L
          ′
        
        →
        0
      
    
    {\displaystyle 0\to I\to L\to L'\to 0}
  
 is split.</li></ul>
2.  An <a href="/facts/Injective_envelope/VoENIwTw">injective envelope</a> (also called injective hull) is a maximal essential extension, or a minimal embedding in an injective module.
3.  An <a href="/facts/Injective_cogenerator/sATNkehI">injective cogenerator</a> is an injective module such that every module has a nonzero homomorphism into it.

invariant
<a href="/facts/Invariant_(mathematics)/9TeNN2ed">invariants</a>

invertible
An <a href="/facts/Invertible_module/apvu7Vmx">invertible module</a> over a commutative ring is a rank-one finite projective module.

irreducible module
Another name for a <a href="/facts/Simple_module/00roKQrG">simple module</a>.

isomorphism
An <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a> between modules is an invertible module homomorphism.

Jacobson
<a href="/facts/Jacobson%2527s_density_theorem/hy5I12Wo">Jacobson's density theorem</a>

Kähler differentials
<a href="/facts/K%25C3%25A4hler_differentials/1aPfygbf">Kähler differentials</a>

Kaplansky
<a href="/facts/Kaplansky%2527s_theorem_on_a_projective_module/LhRhNVbg">Kaplansky's theorem on a projective module</a> says that a projective module over a local ring is free.

kernel
The kernel of a module homomorphism is the pre-image of the zero element.

Koszul complex
<a href="/facts/Koszul_complex/G8N1ugQK">Koszul complex</a>

Krull–Schmidt
The <a href="/facts/Krull%25E2%2580%2593Schmidt_theorem/DWRwHXpU">Krull–Schmidt theorem</a> says that (1) a finite-length module admits an indecomposable decomposition and (2) any two indecomposable decompositions of it are equivalent.

length
The <a href="/facts/Length_of_a_module/UpWodT4k">length of a module</a> is the common length of any composition series of the module; the length is infinite if there is no composition series. Over a field, the length is more commonly known as the <a href="/facts/Dimension_of_a_vector_space/z8m02A3W">dimension</a>.

linear
1.  A linear map is another term for a <a href="/facts/Module_homomorphism/SCSzvgxx">module homomorphism</a>.
2.  <a href="/facts/Linear_topology/5w5TpKx9">Linear topology</a>

localization
<a href="/facts/Localization_of_a_module/rjevzBJf">Localization of a module</a> converts <i>R</i> modules to <i>S</i> modules, where <i>S</i> is a <a href="/facts/Localization_of_a_ring/rjevzBJf">localization</a> of <i>R</i>.

Matlis module
<a href="/facts/Matlis_module/ndBIsJC8">Matlis module</a>

Mitchell's embedding theorem
<a href="/facts/Mitchell%2527s_embedding_theorem/6Mmx0k9e">Mitchell's embedding theorem</a>

Mittag-Leffler
<a href="/facts/Mittag-Leffler_condition/f6QecRUu">Mittag-Leffler condition</a> (ML)

module
1.  A <a href="/facts/Left_module/jP0LIhFL">left module</a> 
  
    
      
        M
      
    
    {\displaystyle M}
  
 over the <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> 
  
    
      
        R
      
    
    {\displaystyle R}
  
 is an <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a> 
  
    
      
        (
        M
        ,
        +
        )
      
    
    {\displaystyle (M,+)}
  
 with an operation 
  
    
      
        R
        ×
        M
        →
        M
      
    
    {\displaystyle R\times M\to M}
  
 (called scalar multipliction) satisfies the following condition:

∀
        r
        ,
        s
        ∈
        R
        ,
        ∀
        m
        ,
        n
        ∈
        M
      
    
    {\displaystyle \forall r,s\in R,\forall m,n\in M}
  
,
<ol><li>
  
    
      
        r
        (
        m
        +
        n
        )
        =
        r
        m
        +
        r
        n
      
    
    {\displaystyle r(m+n)=rm+rn}
  
</li>
<li>
  
    
      
        r
        (
        s
        m
        )
        =
        (
        r
        s
        )
        m
      
    
    {\displaystyle r(sm)=(rs)m}
  
</li>
<li>
  
    
      
        
          1
          
            R
          
        
        
        m
        =
        m
      
    
    {\displaystyle 1_{R}\,m=m}
  
</li></ol>
2.  A <a href="/facts/Right_module/jP0LIhFL">right module</a> 
  
    
      
        M
      
    
    {\displaystyle M}
  
 over the ring 
  
    
      
        R
      
    
    {\displaystyle R}
  
 is an abelian group 
  
    
      
        (
        M
        ,
        +
        )
      
    
    {\displaystyle (M,+)}
  
 with an operation 
  
    
      
        M
        ×
        R
        →
        M
      
    
    {\displaystyle M\times R\to M}
  
 satisfies the following condition:

∀
        r
        ,
        s
        ∈
        R
        ,
        ∀
        m
        ,
        n
        ∈
        M
      
    
    {\displaystyle \forall r,s\in R,\forall m,n\in M}
  
,
<ol><li>
  
    
      
        (
        m
        +
        n
        )
        r
        =
        m
        r
        +
        n
        r
      
    
    {\displaystyle (m+n)r=mr+nr}
  
</li>
<li>
  
    
      
        (
        m
        s
        )
        r
        =
        r
        (
        s
        m
        )
      
    
    {\displaystyle (ms)r=r(sm)}
  
</li>
<li>
  
    
      
        m
        
          1
          
            R
          
        
        =
        m
      
    
    {\displaystyle m1_{R}=m}
  
</li></ol>
3.  All the modules together with all the module homomorphisms between them form the <a href="/facts/Category_of_modules/AnnF9JKf">category of modules</a>.

module spectrum
A <a href="/facts/Module_spectrum/569pLk5l">module spectrum</a> is a <a href="/facts/Spectrum_(topology)/mWho3roy">spectrum</a> with an action of a ring spectrum.

nilpotent
A <a href="/facts/Nilpotent_endomorphism/qIns3FJJ">nilpotent endomorphism</a> is an endomorphism, some power of which is zero.

Noetherian
A <a href="/facts/Noetherian_module/kkxeu4Md">Noetherian module</a> is a module such that every submodule is finitely generated. Equivalently, every increasing chain of submodules becomes stationary after finitely many steps.

normal
normal forms for matrices

perfect
1.  <a href="/facts/Perfect_complex/RAuwLe32">perfect complex</a>
2.  <a href="/facts/Perfect_module/RAuwLe32">perfect module</a>

principal
A <a href="/facts/Principal_indecomposable_module/bpxrA7xs">principal indecomposable module</a> is a cyclic indecomposable projective module.

primary
<a href="/facts/Primary_submodule/dgPs3BYw">primary submodule</a>

projective
A 
  
    
      
        R
      
    
    {\displaystyle R}
  
-module 
  
    
      
        P
      
    
    {\displaystyle P}
  
 is called a <a href="/facts/Projective_module/lHymqhNC">projective module</a> if given a 
  
    
      
        R
      
    
    {\displaystyle R}
  
-module homomorphism 
  
    
      
        g
        :
        P
        →
        M
      
    
    {\displaystyle g:P\to M}
  
, and a <a href="/facts/Surjective/KezhLpCb">surjective</a> 
  
    
      
        R
      
    
    {\displaystyle R}
  
-module homomorphism 
  
    
      
        f
        :
        N
        →
        M
      
    
    {\displaystyle f:N\to M}
  
, there exists a 
  
    
      
        R
      
    
    {\displaystyle R}
  
-module homomorphism 
  
    
      
        h
        :
        P
        →
        N
      
    
    {\displaystyle h:P\to N}
  
 such that 
  
    
      
        f
        ∘
        h
        =
        g
      
    
    {\displaystyle f\circ h=g}
  
.
The following conditions are equivalent:
<ul><li>The covariant functor 
  
    
      
        
          
            
              Hom
            
          
          
            R
          
        
        (
        P
        ,
        −
        )
      
    
    {\displaystyle {\textrm {Hom}}_{R}(P,-)}
  
 is <a href="/facts/Exact_functor/4tsQzwBp">exact</a>.</li>
<li>
  
    
      
        M
      
    
    {\displaystyle M}
  
 is a projective module.</li>
<li>Every short exact sequence 
  
    
      
        0
        →
        L
        →
        
          L
          ′
        
        →
        P
        →
        0
      
    
    {\displaystyle 0\to L\to L'\to P\to 0}
  
 is split.</li>
<li>
  
    
      
        M
      
    
    {\displaystyle M}
  
 is a direct summand of free modules.</li></ul>
In particular, every free module is projective.
2.  The <a href="/facts/Projective_dimension/lHymqhNC">projective dimension</a> of a module is the minimal length of (if any) a finite projective resolution of the module; the dimension is infinite if there is no finite projective resolution.
3.  A <a href="/facts/Projective_cover/spGV7vUQ">projective cover</a> is a minimal surjection from a projective module.

pure submodule
<a href="/facts/Pure_submodule/hv06k1c3">pure submodule</a>

Quillen–Suslin theorem
The <a href="/facts/Quillen%25E2%2580%2593Suslin_theorem/J5o9GfMX">Quillen–Suslin theorem</a> states that a finite projective module over a polynomial ring is free.

quotient
Given a left 
  
    
      
        R
      
    
    {\displaystyle R}
  
-module 
  
    
      
        M
      
    
    {\displaystyle M}
  
 and a submodule 
  
    
      
        N
      
    
    {\displaystyle N}
  
, the <a href="/facts/Quotient_group/HlwpHtjb">quotient group</a> 
  
    
      
        M
        
          /
        
        N
      
    
    {\displaystyle M/N}
  
 can be made to be a left 
  
    
      
        R
      
    
    {\displaystyle R}
  
-module by 
  
    
      
        r
        (
        m
        +
        N
        )
        =
        r
        m
        +
        N
      
    
    {\displaystyle r(m+N)=rm+N}
  
 for 
  
    
      
        r
        ∈
        R
        ,
        
        m
        ∈
        M
      
    
    {\displaystyle r\in R,\,m\in M}
  
. It is called a <a href="/facts/Quotient_module/IJFG2yqv">quotient module</a> or factor module.

radical
The <a href="/facts/Radical_of_a_module/Yz6jStC0">radical of a module</a> is the intersection of the maximal submodules.  For Artinian modules, the smallest submodule with semisimple quotient.

rational
<a href="/facts/Rational_canonical_form/EUPch5SM">rational canonical form</a>

reflexive
A <a href="/facts/Reflexive_module/1tx7kXMg">reflexive module</a> is a module that is isomorphic via the natural map to its second dual.

resolution
<a href="/facts/Resolution_(algebra)/xEFm83U7">resolution</a>

restriction
<a href="/facts/Restriction_of_scalars/RYGsTsYk">Restriction of scalars</a> uses a ring homomorphism from <i>R</i> to <i>S</i> to convert <i>S</i>-modules to <i>R</i>-modules.

Schanuel
<a href="/facts/Schanuel%2527s_lemma/0cCdcjCe">Schanuel's lemma</a>

Schur
<a href="/facts/Schur%2527s_lemma/JvAT7t09">Schur's lemma</a> says that the endomorphism ring of a simple module is a division ring.

Shapiro
<a href="/facts/Shapiro%2527s_lemma/fj6dxd8I">Shapiro's lemma</a>

sheaf of modules
<a href="/facts/Sheaf_of_modules/grSMBXho">sheaf of modules</a>

snake
<a href="/facts/Snake_lemma/UbVGa0Bn">snake lemma</a>

socle
The <a href="/facts/Socle_(mathematics)/TIcneKjK">socle</a> is the largest semisimple submodule.

semisimple
A <a href="/facts/Semisimple_module/9KASYk35">semisimple module</a> is a direct sum of simple modules.

simple
A <a href="/facts/Simple_module/00roKQrG">simple module</a> is a nonzero module whose only submodules are zero and itself.

Smith
<a href="/facts/Smith_normal_form/pchMkW7Q">Smith normal form</a>

stably free
A <a href="/facts/Stably_free_module/uXc6TazR">stably free module</a>

structure theorem
The <a href="/facts/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain/VydAyMF1">structure theorem for finitely generated modules over a principal ideal domain</a> says that a finitely generated modules over PIDs are finite direct sums of primary cyclic modules.

submodule
Given a 
  
    
      
        R
      
    
    {\displaystyle R}
  
-module 
  
    
      
        M
      
    
    {\displaystyle M}
  
, an additive subgroup 
  
    
      
        N
      
    
    {\displaystyle N}
  
 of 
  
    
      
        M
      
    
    {\displaystyle M}
  
 is a <a href="/facts/Submodule/jP0LIhFL">submodule</a> if 
  
    
      
        R
        N
        ⊂
        N
      
    
    {\displaystyle RN\subset N}
  
.

support
The <a href="/facts/Support_of_a_module/GzwFaoTw">support of a module</a> over a commutative ring is the set of prime ideals at which the localizations of the module are nonzero.

tensor
<a href="/facts/Tensor_product_of_modules/K3oaFIS4">Tensor product of modules</a>

topological
A <a href="/facts/Topological_module/AXASMgd6">topological module</a>

Tor
<a href="/facts/Tor_functor/oFn2L3gd">Tor functor</a>

torsion-free
<a href="/facts/Torsion-free_module/IFsk1xw0">torsion-free module</a>

torsionless
<a href="/facts/Torsionless_module/1tx7kXMg">torsionless module</a>

uniform
A <a href="/facts/Uniform_module/pK7y9SGd">uniform module</a> is a module in which every two non-zero submodules have a non-zero intersection.

weak
<a href="/facts/Weak_dimension/ggvBlFhU">weak dimension</a>

zero
1.  The <a href="/facts/Zero_module/hNAAkeen">zero module</a> is a module consisting of only zero element.
2.  The <a href="/facts/Zero_module_homomorphism/4NBRtKHc">zero module homomorphism</a> is a module homomorphism that maps every element to zero.

<ul><li>John A. Beachy (1999). <a href="https://archive.org/details/introductorylect0000beac"><i>Introductory Lectures on Rings and Modules</i></a> (1st ed.). <a href="/facts/Addison-Wesley/A7UFATEG">Addison-Wesley</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-64407-0.</li>
<li>Golan, Jonathan S.; Head, Tom (1991), <a href="https://archive.org/details/modulesstructure0000gola"><i>Modules and the structure of rings</i></a>, Monographs and Textbooks in Pure and Applied Mathematics, vol. 147, Marcel Dekker, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8247-8555-0, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1201818">1201818</a></li>
<li>Lam, Tsit-Yuen (1999), <i>Lectures on modules and rings</i>, Graduate Texts in Mathematics No. 189, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-98428-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1653294">1653294</a></li>
<li><a href="/facts/Serge_Lang/I7MW2rUu">Serge Lang</a> (1993). <i>Algebra</i> (3rd ed.). <a href="/facts/Addison-Wesley/A7UFATEG">Addison-Wesley</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-201-55540-9.</li>
<li>Passman, Donald S. (1991), <i>A course in ring theory</i>, The Wadsworth & Brooks/Cole Mathematics Series, Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-534-13776-2, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1096302">1096302</a></li></ul>

Glossary of module theory open-in-new

Glossary of module theory