algebra ideal

For an algebra , a subset is called an ideal if

1. is a vector subspace of
2. Closed under multiplication: for all .
3. (technically this condition means is a left ideal)

4. (technically this condition means is a right ideal)

generating ideals

For a subset , the ideal of generated by is

We can think of it as the smallest ideal that contains all of . If is not an ideal, it will have other stuff in it, but this is the “best” ideal with as little other stuff as possible.

Explicit construction

is all finite -linear combos of elements of .

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References