Algebra theory MOC

Algebra ideal

A subalgebra¹ 𝐼 ≤𝐴 of an algebra 𝐴 over a field 𝕂 is called a left-ideal iff 𝐴𝐼 ⊆𝐼, a right-ideal iff 𝐼𝐴 ⊆𝐼, and a two-sided ideal (sometimes just ideal) iff both conditions hold,² falg i.e. a left-ideal absorbs elements placed on the left, &c. Compare with an ideal of a rng. Given a two-sided ideal one may construct a Quotient algebra.

Special cases

• Lie algebra ideal

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develop | en | SemBr

Footnotes

1. Note that any vector subspace satisfying the definition is automatically a subalgebra. ↩

2. 1988. Vertex operator algebras and the Monster, §1.3, p. 6 ↩