Proving if k-algebras are finitely generated and noetherian
Given two $k$-algebras $R:=koplus x^2 , k[x] subseteq k[x]$ and $S:= koplus xy ,, k[x,y] subseteq k[x,y]$ I need to show if they are finitely generated and if they are noetherian.
For $R$ to be finitely generated k-Algebra, we need to show that there exist finitely many elements, $a_1, a_2,....a_n$ such that $R=k[a_1,a_2,...a_n].$ But we know that it is contained in $k[x]$. For the second Question I Need to show if there is an ascending chain of ideals.
Can somebody give me a hint as how to proceed ? Many thanks.
abstract-algebra
asked Nov 25 at 14:15
user249018
338127
add a comment |
Given two $k$-algebras $R:=koplus x^2 , k[x] subseteq k[x]$ and $S:= koplus xy ,, k[x,y] subseteq k[x,y]$ I need to show if they are finitely generated and if they are noetherian.
For $R$ to be finitely generated k-Algebra, we need to show that there exist finitely many elements, $a_1, a_2,....a_n$ such that $R=k[a_1,a_2,...a_n].$ But we know that it is contained in $k[x]$. For the second Question I Need to show if there is an ascending chain of ideals.
Can somebody give me a hint as how to proceed ? Many thanks.
abstract-algebra
asked Nov 25 at 14:15
user249018
338127
add a comment |
Given two $k$-algebras $R:=koplus x^2 , k[x] subseteq k[x]$ and $S:= koplus xy ,, k[x,y] subseteq k[x,y]$ I need to show if they are finitely generated and if they are noetherian.
For $R$ to be finitely generated k-Algebra, we need to show that there exist finitely many elements, $a_1, a_2,....a_n$ such that $R=k[a_1,a_2,...a_n].$ But we know that it is contained in $k[x]$. For the second Question I Need to show if there is an ascending chain of ideals.
Can somebody give me a hint as how to proceed ? Many thanks.
abstract-algebra
asked Nov 25 at 14:15
user249018
338127
Given two $k$-algebras $R:=koplus x^2 , k[x] subseteq k[x]$ and $S:= koplus xy ,, k[x,y] subseteq k[x,y]$ I need to show if they are finitely generated and if they are noetherian.
For $R$ to be finitely generated k-Algebra, we need to show that there exist finitely many elements, $a_1, a_2,....a_n$ such that $R=k[a_1,a_2,...a_n].$ But we know that it is contained in $k[x]$. For the second Question I Need to show if there is an ascending chain of ideals.
Can somebody give me a hint as how to proceed ? Many thanks.
abstract-algebra
abstract-algebra
asked Nov 25 at 14:15
user249018
338127
asked Nov 25 at 14:15
user249018
338127
asked Nov 25 at 14:15
user249018
338127
asked Nov 25 at 14:15
user249018
338127
asked Nov 25 at 14:15
user249018
338127
338127
add a comment |
add a comment |
1 Answer
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$R = k oplus x^2 k[x] = k [ x^2, x^3 ]$
Also, $S$ is isomorphic to the semi-group algebra $k [ mathbb N times mathbb N ]$, which is not a finitely generated semi-group.
None of the $(n, 1)$ can be written in terms of others. So $S$ is not Noetherian.
answered Nov 25 at 15:39
R.C.Cowsik
31514
add a comment |
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1 Answer
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1 Answer
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$R = k oplus x^2 k[x] = k [ x^2, x^3 ]$
Also, $S$ is isomorphic to the semi-group algebra $k [ mathbb N times mathbb N ]$, which is not a finitely generated semi-group.
None of the $(n, 1)$ can be written in terms of others. So $S$ is not Noetherian.
answered Nov 25 at 15:39
R.C.Cowsik
31514
add a comment |
$R = k oplus x^2 k[x] = k [ x^2, x^3 ]$
Also, $S$ is isomorphic to the semi-group algebra $k [ mathbb N times mathbb N ]$, which is not a finitely generated semi-group.
None of the $(n, 1)$ can be written in terms of others. So $S$ is not Noetherian.
answered Nov 25 at 15:39
R.C.Cowsik
31514
add a comment |
$R = k oplus x^2 k[x] = k [ x^2, x^3 ]$
Also, $S$ is isomorphic to the semi-group algebra $k [ mathbb N times mathbb N ]$, which is not a finitely generated semi-group.
None of the $(n, 1)$ can be written in terms of others. So $S$ is not Noetherian.
answered Nov 25 at 15:39
R.C.Cowsik
31514
$R = k oplus x^2 k[x] = k [ x^2, x^3 ]$
Also, $S$ is isomorphic to the semi-group algebra $k [ mathbb N times mathbb N ]$, which is not a finitely generated semi-group.
None of the $(n, 1)$ can be written in terms of others. So $S$ is not Noetherian.
answered Nov 25 at 15:39
R.C.Cowsik
31514
answered Nov 25 at 15:39
R.C.Cowsik
31514
answered Nov 25 at 15:39
R.C.Cowsik
31514
answered Nov 25 at 15:39
R.C.Cowsik
31514
31514
add a comment |
add a comment |
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