Algebra theory MOC

Algebra ideal

A $K$ -submodule¹ $𝐼 \leq 𝐴$ of a $K$ -algebra called a left-ideal iff $𝐴 𝐼 \subseteq 𝐼$ , a right-ideal iff $𝐼 𝐴 \subseteq 𝐼$ , and a two-sided ideal (sometimes just ideal) iff both conditions hold,² #m/def/ralg i.e. a left-ideal absorbs elements placed on the left, &c.

For an associatie algebra (in particular $K$ -monoid) ideal of a rng.
Given a two-sided ideal $𝐼$ one may construct a quotient algebra $𝐴 / 𝐼$ .

Special cases

Lie algebra ideal

#state/tidy | #lang/en | #SemBr

Note that any vector subspace satisfying the definition is automatically a subalgebra. ↩
1988. Vertex operator algebras and the Monster, §1.3, p. 6 ↩