Ideal of a ring

Definition
Suppose $$R$$ is a non-associative ring (i.e., a ring that is not necessarily commutative, associative, unital or Lie). A subset $$I$$ of $$R$$ is termed an ideal (or two-sided ideal) of $$R$$ if it satisfies the following two conditions:


 * 1) $$I$$ is an additive subgroup of the additive group of $$R$$.
 * 2) For any $$x \in I$$ and $$y \in R$$, both $$x * y$$ and $$y * x$$ are in $$I$$, where $$*$$ is the multiplication operation of $$R$$.

Related notions

 * Ideal of a Lie ring
 * Left ideal of a ring
 * Right ideal of a ring
 * Ideal of a cring