Example

(1) Let $A$ be a ring. The set $A$ is a two-sided ideal of $A$; so is the set consisting of 0, which is called the zero ideal and sometimes denoted by 0 or $(0)$ instead of $\{0\}$.

(2) For every element $a$ of $A$, the set $A.a$ of left multiples of $a$ is a left ideal; similarly the set $a.A$ is a right ideal. When $a$ is in the centre of $A$, $A.a = a.A$; this ideal is called the principal ideal generated by $a$ and is denoted by $(a)$. $(a) = A$ if and only if $a$ is invertible.

(3) Let $M$ be a subset of $A$. The set of elements $x \in A$ such that $xy = 0$ for all $y \in M$ is a left ideal of $A$ called the left annihilator of $M$. The right annihilator of $M$ is defined similarly.

(4) Every intersection of left (resp. right, two-sided) of $A$ is a left (resp. right, two-sided) ideal. Given a subset $X$ of $A$, there thus exists a smallest left (resp. right, two-sided) ideal containing $X$; it is called the left (resp. right, two-sided) ideal generated by $X$.

Let $a$ be a left ideal of $A$. The conditions $1 \notin a, a \neq A$ are obviously equivalent.