Ideal

definitionring-theory

Ring ideals are the analogue of normal subgroups.

Definition

For ring $(R,+,\cdot )$, and $I\le R$ a subring (not usually unital), we say that $I$ is a left ideal (denoted $I⊴R$) if

$$\forall x\in I,\forall r\in R,rx\in I,$$

$I$ is a right ideal if

$$\forall x\in I,\forall r\in R,xr\in I,$$

or $I$ is a two sided ideal if

$$\forall x\in I,\forall r\in R,rx,xr\in I.$$

If $R$ is commutative, these are all equivalent and we can simply call $I$ an ideal.

Examples