radical of an ideal

Let R be a commutative ring. For any ideal I of R, the radicalPlanetmathPlanetmath of I, written I or Rad(I), is the set

{aRanI for some integer n>0}

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

If I=I, then I is called a radical ideal.

Every prime idealMathworldPlanetmathPlanetmathPlanetmath is a radical ideal. If I is a radical ideal, the quotient ringMathworldPlanetmath R/I is a ring with no nonzero nilpotent elementsMathworldPlanetmath.

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 aR such that every m-system containing a has a non-empty intersectionMathworldPlanetmath with I:

I:={aRif S is an m-system, and aS, then SI}.

Under this definition, we see that I is again an ideal (two-sided) and it is a subset of {aRanI for some integer n>0}. Furthermore, if R is commutativePlanetmathPlanetmathPlanetmath, the two sets coincide. In other words, this definition of a radical of an ideal is indeed a “generalizationPlanetmathPlanetmath” 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