# radical of an ideal

Let $R$ be a commutative ring. For any ideal $I$ of $R$, the radical^{} of $I$, written $\sqrt{I}$ or $\mathrm{Rad}(I)$, is the set

$$\{a\in R\mid {a}^{n}\in I\text{for some integer}n0\}$$ |

The radical of an ideal $I$ is always an ideal of $R$.

If $I=\sqrt{I}$, then $I$ is called a radical ideal.

Every prime ideal^{} is a radical ideal. If $I$ is a radical ideal, the quotient ring^{} $R/I$ is a ring with no nonzero nilpotent elements^{}.

More generally, the radical of an ideal in can be defined over an arbitrary ring. Let $I$ be an ideal of a ring $R$, the radical of $I$ is the set of $a\in R$ such that every m-system containing $a$ has a non-empty intersection^{} with $I$:

$$\sqrt{I}:=\{a\in R\mid \text{if}S\text{is an}m\text{-system, and}a\in S,\text{then}S\cap I\ne \mathrm{\varnothing}\}.$$ |

Under this definition, we see that $\sqrt{I}$ is again an ideal (two-sided) and it is a subset of $\{a\in R\mid {a}^{n}\in I\text{for some integer}n0\}$. Furthermore, if $R$ is commutative^{}, the two sets coincide. In other words, this definition of a radical of an ideal is indeed a “generalization^{}” of the radical of an ideal in a commutative ring.

Title | radical of an ideal |

Canonical name | RadicalOfAnIdeal |

Date of creation | 2013-03-22 12:35:54 |

Last modified on | 2013-03-22 12:35:54 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 17 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 14A05 |

Classification | msc 16N40 |

Classification | msc 13-00 |

Related topic | PrimeRadical |

Related topic | RadicalOfAnInteger |

Related topic | JacobsonRadical |

Related topic | HilbertsNullstellensatz |

Related topic | AlgebraicSetsAndPolynomialIdeals |

Defines | radical ideal |

Defines | radical |