finitely generated algebra

Definition

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

Example: is finitely generated by .

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 and the subalgebra . this subalgebra is not finitely generated.

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References