Let $S$ be a semigroup. An ideal of $S$ is a non-empty subset of $S$ which is closed under multiplication on either side by elements of $S$ . Formally, $I$ is an ideal of $S$ if $I$ is non-empty, and for all $x \in I$ and $s \in S$ , we have $sx \in I$ and $xs \in I$ .

One-sided ideals are defined similarly. A non-empty subset $A$ of $S$ is a left ideal (resp. right ideal) of $S$ if for all $a \in A$ and $s \in S$ , we have $sa \in A$ (resp. $as \in A$ ).

A principal left ideal of $S$ is a left ideal generated by a single element. If $a \in S$ , then the principal left ideal of $S$ generated by $a$ is $S^1a = Sa \cup \{a\}$ . (The notation $S^1$ is explained here.)

Similarly, the principal right ideal generated by $a$ is $aS^1 = aS \cup \{a\}$ .

The notation $L(a)$ and $R(a)$ are also common for the principal left and right ideals generated by $a$ respectively.

A principal ideal of $S$ is an ideal generated by a single element. The ideal generated by $a$ is $$S^1aS^1 = SaS \cup Sa \cup aS \cup \{a\}.$$ The notation $J(a) = S^1aS^1$ is also common.