[[Algebra theory MOC]]
# Algebra ideal

A [[Subalgebra over a field|subalgebra]][^subs] $I \leq A$ of an [[K-algebra|algebra]] $A$ over a field $\mathbb{K}$ is called a **left-ideal** iff $AI \sube I$,
a **right-ideal** iff $IA \sube I$,
and a **two-sided ideal** (sometimes just **ideal**) iff both conditions hold,[^1988] #m/def/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]].

  [^subs]: Note that any vector subspace satisfying the definition is automatically a subalgebra.


  [^1988]: 1988\. [[Sources/@frenkelVertexOperatorAlgebras1988|Vertex operator algebras and the Monster]], §1.3, p. 6

## Special cases

- [[Lie algebra ideal]]

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