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.