# 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 (http://planetmath.org/AdjoiningAnIdentityToASemigroup3).)

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

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

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

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

Title | ideal |

Canonical name | Ideal1 |

Date of creation | 2013-03-22 13:05:43 |

Last modified on | 2013-03-22 13:05:43 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 8 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 20M12 |

Classification | msc 20M10 |

Related topic | ReesFactor |

Defines | left ideal |

Defines | right ideal |

Defines | principal ideal |

Defines | principal left ideal |

Defines | principal right ideal |