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Is $\mathbb{C}[x,y]$ finitely generated $\mathbb{C}$-algebra? Also is it 2-generated?

As I can't see the reason why this is true, yet we are using reasoning like this in a course in non commutative algebra. If something is finitely generated, then it's automatically notherian.

Got a matrix2x2 of $\mathbb{C}[x,y]$ and need to show that it's notherian.

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    @MarianoSuárez-Alvarez all I seem to be doing is studying this and still don't understand it. Anyway this has cleared things up a lot.2011-11-16

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As noted in comments, finitely-generated is not the same as finite-dimensional.

See http://en.wikipedia.org/wiki/Finitely-generated_algebra for definition of "finitely generated."

This extends the notion of "finitely generated group," for if $G$ is a group and $K$ a field, the algebra $K[G]$ is finitely generated as a $K$-algebra if and only if $G$ is a finitely-generated group.

Now, $\mathbb C[x,y]$ is finitely-generated almost by definition - every element of the algebra is a represented as a polynomial in $x$ and $y$, and hence it is generated by 2 elements.

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    I've seen people use $K\langle x, y\rangle$ for something like that.2011-11-16