Generating set of a module

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a generating set Γ of a <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> M over a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> R is a <a href="/facts/Subset/SJGERmbA">subset</a> of M such that the smallest <a href="/facts/Submodule/jP0LIhFL">submodule</a> of M containing Γ is M itself (the smallest submodule containing a subset is the <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersection</a> of all submodules containing the set). The set Γ is then said to generate M. For example, the ring R is generated by the identity element 1 as a left R-module over itself. If there is a <a href="/facts/Finite_set/za1newlP">finite</a> generating set, then a module is said to be <a href="/facts/Finitely_generated_module/6Njcpgzr">finitely generated</a>.
This applies to <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideals</a>, which are the submodules of the ring itself. In particular, a <a href="/facts/Principal_ideal/Cga66qgM">principal ideal</a> is an ideal that has a generating set consisting of a single element.
Explicitly, if Γ is a generating set of a module M, then every element of M is a (finite) R-linear combination of some elements of Γ; i.e., for each x in M, there are r1, ..., rm in R and g1, ..., gm in Γ such that

x
        =
        
          r
          
            1
          
        
        
          g
          
            1
          
        
        +
        ⋯
        +
        
          r
          
            m
          
        
        
          g
          
            m
          
        
        .
      
    
    {\displaystyle x=r_{1}g_{1}+\cdots +r_{m}g_{m}.}

Put in another way, there is a <a href="/facts/Surjection/KezhLpCb">surjection</a>

⨁
          
            g
            ∈
            Γ
          
        
        R
        →
        M
        ,
        
        
          r
          
            g
          
        
        ↦
        
          r
          
            g
          
        
        g
        ,
      
    
    {\displaystyle \bigoplus _{g\in \Gamma }R\to M,\,r_{g}\mapsto r_{g}g,}

where we wrote rg for an element in the g-th component of the direct sum. (Coincidentally, since a generating set always exists, e.g. M itself, this shows that a module is a <a href="/facts/Quotient_module/IJFG2yqv">quotient</a> of a <a href="/facts/Free_module/o6rC6yoJ">free module</a>, a useful fact.)
A generating set of a module is said to be minimal if no <a href="/facts/Proper_subset/SJGERmbA">proper subset</a> of the set generates the module. If R is a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>, then a minimal generating set is the same thing as a <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a>. Unless the module is <a href="/facts/Finitely_generated_module/6Njcpgzr">finitely generated</a>, there may exist no minimal generating set.
The <a href="/facts/Cardinality/m6VPojwT">cardinality</a> of a minimal generating set need not be an invariant of the module; Z is generated as a principal ideal by 1, but it is also generated by, say, a minimal generating set {2, 3}. What is uniquely determined by a module is the <a href="/facts/Infimum/oXCKVopz">infimum</a> of the numbers of the generators of the module.
Let R be a <a href="/facts/Local_ring/gRTdg5Qx">local ring</a> with <a href="/facts/Maximal_ideal/nf4DnKVS">maximal ideal</a> m and <a href="/facts/Residue_field/e6nflEmB">residue field</a> k and M finitely generated module. Then <a href="/facts/Nakayama%2527s_lemma/Nwhck7ru">Nakayama's lemma</a> says that M has a minimal generating set whose cardinality is 
 
 
 
 
 dim
 
 k
 
 
 ⁡
 M
 
 /
 
 m
 M
 =
 
 dim
 
 k
 
 
 ⁡
 M
 
 ⊗
 
 R
 
 
 k
 
 
 {\displaystyle \dim _{k}M/mM=\dim _{k}M\otimes _{R}k}
 
. If M is <a href="/facts/Flat_module/1WFrnJP3">flat</a>, then this minimal generating set is <a href="/facts/Linearly_independent/8JrHfIIa">linearly independent</a> (so M is free). See also: <a href="/facts/Minimal_resolution_(algebra)/xEFm83U7">Minimal resolution</a>.
A more refined information is obtained if one considers the relations between the generators; see <a href="/facts/Free_presentation_of_a_module/GxxKGm4L">Free presentation of a module</a>.

Generating set of a module open-in-new

Generating set of a module