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# ideal

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 $S^{1}a=Sa\cup\{a\}$. (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

$S^{1}aS^{1}=SaS\cup Sa\cup aS\cup\{a\}.$ |

The notation $J(a)=S^{1}aS^{1}$ is also common.

## Mathematics Subject Classification

20M12*no label found*20M10

*no label found*

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