finitely generated algebra

We call an algebra $A$ finitely generated if $A=(S)$ for a finite subset $S$. This is important, because it tells us how to “build” the algebra from just a finite set of pieces.

Example: $\mathbb{C}[x_1,\dots ,x_n]$ is finitely generated by $x_1,\dots ,x_n$.

It is interesting to note that unlike dimension, which plays nice with subspaces, we can find a subalgebra of a finitely generated algebra that is not finitely generated.

For example, consider $\mathbb{C}[x,y]$ and the subalgebra $\mathbb{C}[xy,x{y}^2,x{y}^3,\dots ]$. this subalgebra is not finitely generated.

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References