Let say you have a ring $R={\left(\begin{array}{cccc} \mathbb{C}[x,y] & y^2\mathbb{C}[x,y]\\ xy\mathbb{C}[x,y] & \mathbb{C}[x,y] \end{array} \right)}$
It it enough to prove that it's a subring of $M_2(\mathbb{C}[x,y])$ as then from that being a finitely generated $\mathbb{C}$-algebra I.e. it's generated by 8 elements(Is it generated by 8 elements?). Isn't it automatic that the subring is noetherian because all finitely generated rings are Noetherian.
Also, it is true that a subring is finitely generated if the ring is finitely generated?
I fear it isn't enough. However, in my notes he makes weird arguments like this. In proof that
$R={\left(\begin{array}{cccc} \mathbb{Z} & 3\mathbb{Z}\\ \mathbb{Z} & \mathbb{Z} \end{array} \right)}$ is Noetherian.
He says that it contains the ring of scalars which is isomorphic to $\mathbb{Z}$ over which is 4 generated. From this he concludes it is Noetherian.