algebra ideal

For an algebra $A$, a subset $B\subset A$ is called an ideal if

1. $B$ is a vector subspace of $A$
2. Closed under multiplication: $bc\in B$ for all $b,c\in B$.
3. (technically this condition means $B$ is a left ideal)

$$ab\in B\forall a\in A,b\in B$$

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

$$ba\in B\forall a\in A,b\in B$$

generating ideals

For a subset $S\subset A$, the ideal of $A$ generated by $S$ is

$$⟨S⟩:=\bigcap _{\begin{array}{r}I\subset A\text{ideal}\\ \text{s.t.}S\subset I\end{array}}I$$

We can think of it as the smallest ideal that contains all of $S$. If $S$ 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

$⟨S⟩$ is all finite $A$-linear combos of elements of $S$.

$$⟨S⟩=\left\{\sum _{j=1}^n{a}_j{b}_j|n\in {\mathbb{Z}}^{+},a_j\in A,b_j\in S\right\}$$

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References