radical of an ideal
Let R be a commutative ring. For any ideal I of R, the radical of I, written √I or Rad(I), is the set
{a∈R∣an∈I 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 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∈R such that every m-system containing a has a non-empty intersection with I:
√I:= |
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 |