Finitely generated algebra

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a finitely generated algebra (also called an algebra of finite type) is a <a href="/facts/Commutative_algebra_(structure)/DRfoF6KV">commutative</a> <a href="/facts/Associative_algebra/DRfoF6KV">associative algebra</a> 
 
 
 
 A
 
 
 {\displaystyle A}
 
 over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> 
 
 
 
 K
 
 
 {\displaystyle K}
 
 where there exists a finite set of elements 
 
 
 
 
 a
 
 1
 
 
 ,
 …
 ,
 
 a
 
 n
 
 
 
 
 {\displaystyle a_{1},\dots ,a_{n}}
 
 of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 such that every element of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 can be expressed as a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> in 
 
 
 
 
 a
 
 1
 
 
 ,
 …
 ,
 
 a
 
 n
 
 
 
 
 {\displaystyle a_{1},\dots ,a_{n}}
 
, with <a href="/facts/Coefficient/5Hwq2Wuq">coefficients</a> in 
 
 
 
 K
 
 
 {\displaystyle K}
 
.
Equivalently, there exist elements 
 
 
 
 
 a
 
 1
 
 
 ,
 …
 ,
 
 a
 
 n
 
 
 ∈
 A
 
 
 {\displaystyle a_{1},\dots ,a_{n}\in A}
 
 such that the evaluation homomorphism at 
 
 
 
 
 
 a
 
 
 =
 (
 
 a
 
 1
 
 
 ,
 …
 ,
 
 a
 
 n
 
 
 )
 
 
 {\displaystyle {\bf {a}}=(a_{1},\dots ,a_{n})}

ϕ
          
            
              a
            
          
        
        :
        K
        [
        
          X
          
            1
          
        
        ,
        …
        ,
        
          X
          
            n
          
        
        ]
        ↠
        A
      
    
    {\displaystyle \phi _{\bf {a}}\colon K[X_{1},\dots ,X_{n}]\twoheadrightarrow A}

is <a href="/facts/Surjective/KezhLpCb">surjective</a>; thus, by applying the <a href="/facts/First_isomorphism_theorem/SNpPTkrd">first isomorphism theorem</a>, 
 
 
 
 A
 ≃
 K
 [
 
 X
 
 1
 
 
 ,
 …
 ,
 
 X
 
 n
 
 
 ]
 
 /
 
 
 
 k
 e
 r
 
 
 (
 
 ϕ
 
 
 a
 
 
 
 )
 
 
 {\displaystyle A\simeq K[X_{1},\dots ,X_{n}]/{\rm {ker}}(\phi _{\bf {a}})}
 
. 
<a href="/facts/Converse_(logic)/VhlUrxxx">Conversely</a>, 
 
 
 
 A
 :=
 K
 [
 
 X
 
 1
 
 
 ,
 …
 ,
 
 X
 
 n
 
 
 ]
 
 /
 
 I
 
 
 {\displaystyle A:=K[X_{1},\dots ,X_{n}]/I}
 
 for any <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a> 
 
 
 
 I
 ⊆
 K
 [
 
 X
 
 1
 
 
 ,
 …
 ,
 
 X
 
 n
 
 
 ]
 
 
 {\displaystyle I\subseteq K[X_{1},\dots ,X_{n}]}
 
 is a 
 
 
 
 K
 
 
 {\displaystyle K}
 
-algebra of finite type, indeed any element of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a polynomial in the cosets 
 
 
 
 
 a
 
 i
 
 
 :=
 
 X
 
 i
 
 
 +
 I
 ,
 i
 =
 1
 ,
 …
 ,
 n
 
 
 {\displaystyle a_{i}:=X_{i}+I,i=1,\dots ,n}
 
 with coefficients in 
 
 
 
 K
 
 
 {\displaystyle K}
 
. Therefore, we obtain the following characterisation of finitely generated 
 
 
 
 K
 
 
 {\displaystyle K}
 
-algebras

A
 
 
 {\displaystyle A}
 
 is a finitely generated 
 
 
 
 K
 
 
 {\displaystyle K}
 
-algebra if and only if it is <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> as a 
 
 
 
 K
 
 
 {\displaystyle K}
 
-algebra to a <a href="/facts/Quotient_ring/X2WkXyEw">quotient ring</a> of the type 
 
 
 
 K
 [
 
 X
 
 1
 
 
 ,
 …
 ,
 
 X
 
 n
 
 
 ]
 
 /
 
 I
 
 
 {\displaystyle K[X_{1},\dots ,X_{n}]/I}
 
 by an ideal 
 
 
 
 I
 ⊆
 K
 [
 
 X
 
 1
 
 
 ,
 …
 ,
 
 X
 
 n
 
 
 ]
 
 
 {\displaystyle I\subseteq K[X_{1},\dots ,X_{n}]}
 
.
If it is necessary to emphasize the field K then the algebra is said to be finitely generated over K. Algebras that are not finitely generated are called infinitely generated.

Finitely generated algebra open-in-new

Finitely generated algebra