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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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    "i.e." followed by "also" is confusing English. $\mathbb C[X,Y]$ is a finitely-generated $\mathbb C$-algebra, but it is not finitely generated as a $\mathbb Q$-algebra. And finitely-generated is not the same as finite-dimensional. http://en.wikipedia.org/wiki/Finitely-generated_algebra2011-11-16
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    @ThomasAndrews That where I'm getting confused as hell. Thought finitely generated mean't finite dimensional.2011-11-16
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    Finitely generated as *what*? 2-dimensional as what?2011-11-16
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    Mean't finite generated $\mathbb{C}$- algebra. Was wondering if that mean't it's a finite dimensional vector space. I personally don't understand any of this, hence poorly explained.2011-11-16
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    If you don't understand, ask your instructor, read your notes/textbook, read *another* textbook on the subject (there are many), and/or ask here.2011-11-16
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    Also, please edit the question to include that information, so that people do not have to read all the comments to know what is being asked.2011-11-16
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    Although $\mathbb{C}[x,y]$ is a finitely-generated $\mathbb{C}$-algebra, it isn't a finite dimensional vector space, as the set $\{x^i y^j\,\vert\,i,j\in \mathbb{N}\}$ is linearly independent over $\mathbb{C}$.2011-11-16
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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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    Query to readers: What is the notation for the free (not necessarily commutative) algebra on $x,y$ (or more generally, on $x1,x2,x3,...x_n$) over $K$? It's obviously the non-commutative polynomials with coefficients in $K$, but does it have a notation like the commutative $K[x,y]$?2011-11-16
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    I've seen people use $K\langle x, y\rangle$ for something like that.2011-11-16