Invariant factor

The invariant factors of a <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> over a <a href="/facts/Principal_ideal_domain/JCwAUnyW">principal ideal domain</a> (PID) occur in one form of 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>.
If 
 
 
 
 R
 
 
 {\displaystyle R}
 
 is a <a href="/facts/Principal_ideal_domain/JCwAUnyW">PID</a> and 
 
 
 
 M
 
 
 {\displaystyle M}
 
 a <a href="/facts/Finitely-generated_module/6Njcpgzr">finitely generated</a> 
 
 
 
 R
 
 
 {\displaystyle R}
 
-module, then

M
        ≅
        
          R
          
            r
          
        
        ⊕
        R
        
          /
        
        (
        
          a
          
            1
          
        
        )
        ⊕
        R
        
          /
        
        (
        
          a
          
            2
          
        
        )
        ⊕
        ⋯
        ⊕
        R
        
          /
        
        (
        
          a
          
            m
          
        
        )
      
    
    {\displaystyle M\cong R^{r}\oplus R/(a_{1})\oplus R/(a_{2})\oplus \cdots \oplus R/(a_{m})}

for some integer 
 
 
 
 r
 ≥
 0
 
 
 {\displaystyle r\geq 0}
 
 and a (possibly empty) list of nonzero elements 
 
 
 
 
 a
 
 1
 
 
 ,
 …
 ,
 
 a
 
 m
 
 
 ∈
 R
 
 
 {\displaystyle a_{1},\ldots ,a_{m}\in R}
 
 for which 
 
 
 
 
 a
 
 1
 
 
 ∣
 
 a
 
 2
 
 
 ∣
 ⋯
 ∣
 
 a
 
 m
 
 
 
 
 {\displaystyle a_{1}\mid a_{2}\mid \cdots \mid a_{m}}
 
. The nonnegative integer 
 
 
 
 r
 
 
 {\displaystyle r}
 
 is called the free rank or Betti number of the module 
 
 
 
 M
 
 
 {\displaystyle M}
 
, while 
 
 
 
 
 a
 
 1
 
 
 ,
 …
 ,
 
 a
 
 m
 
 
 
 
 {\displaystyle a_{1},\ldots ,a_{m}}
 
 are the invariant factors of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 and are unique up to <a href="/facts/Associatedness/cylt6zxW">associatedness</a>.
The invariant factors of a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> over a PID occur in the <a href="/facts/Smith_normal_form/pchMkW7Q">Smith normal form</a> and provide a means of computing the structure of a module from a set of generators and relations.

Invariant factor open-in-new

Invariant factor