# ideal generated by a subset of a ring

Let $X$ be a subset of a ring $R$. Let $S=\{{I}_{k}\}$ be the collection of all left ideals^{} of $R$ that contain $X$ (note that the set is nonempty since $X\subset R$ and $R$ is an ideal in itself). The intersection

$I={\displaystyle \bigcap _{{I}_{k}\in S}}{I}_{k}$ |

is called the *left ideal generated by $X$*, and is denoted by $(X)$. We say that $X$ *generates* $I$ as an ideal.

The definition is symmetrical for right ideals.

Alternatively, we can constructively form the set of elements that constitutes this ideal: The left ideal $(X)$ consists of finite $R$-linear combinations^{} of elements of $X$:

$(X)=\{{\displaystyle \sum _{\lambda}}({r}_{\lambda}{a}_{\lambda}+{n}_{\lambda}{a}_{\lambda})\mid {a}_{\lambda}\in X,{r}_{\lambda}\in R,{n}_{\lambda}\in \mathbb{Z}\}.$ |

Title | ideal generated by a subset of a ring |

Canonical name | IdealGeneratedByASubsetOfARing |

Date of creation | 2013-03-22 14:39:04 |

Last modified on | 2013-03-22 14:39:04 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 9 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 16D25 |

Related topic | GeneratorsOfInverseIdeal |

Related topic | PrimeIdealsByKrullArePrimeIdeals |

Defines | ideal generated by |

Defines | left ideal generated by |

Defines | right ideal generated by |

Defines | generate as an ideal |

Defines | generates as an ideal |

Defines | generates |