radical of an ideal
The radical of an ideal is always an ideal of .
If , then is called a radical ideal.
Every prime ideal is a radical ideal. If is a radical ideal, the quotient ring is a ring with no nonzero nilpotent elements.
More generally, the radical of an ideal in can be defined over an arbitrary ring. Let be an ideal of a ring , the radical of is the set of such that every m-system containing has a non-empty intersection with :
Under this definition, we see that is again an ideal (two-sided) and it is a subset of . Furthermore, if 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 |