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

$$\{a\in R\mid a^n\in I\text{ for some integer }n>0\}$$

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 $\{a\in R\mid a^n\in I\text{ for some integer }n>0\}$. 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