Definition 4

Let $A$ be a ring. A subset $a$ of $A$ is called a left (resp. right) ideal if it is a subgroup of the additive group of $A$ and the relations $a \in A, x \in a$ imply $ax \in a$ (resp. $xa \in a$). $a$ is called a two-sided ideal of $A$ if it is both a left ideal and a right ideal of $A$.

The definition of a left ideal may be expressed by the relations

the relation $-a \subset a$ following from the formula $(-1).x = -x$ and $A.a \subset a$. For all $x \in A$, let $\gamma_x$ be the mapping $a \mapsto xa$ of $A$ into $A$; the action $x \mapsto \gamma_x$ gives the additive group $A^+$ of $A$ the structure of a group with operators with $A$ as set of operators. The left ideals of $A$ are just the subgroups of $A^+$ which are stable under this action.

The left ideals in the ring $A$ are just the right ideals of the opposite ring $A^0$. In a commutative ring the three species of ideals are the same; they are simply called ideals.