Jacobson radical
The Jacobson radical of a unital ring is the intersection
of the annihilators
of simple (http://planetmath.org/SimpleModule) left -modules.
The following are alternative characterizations of the Jacobson radical :
-
1.
The intersection of all left primitive ideals.
-
2.
The intersection of all maximal left ideals
.
-
3.
The set of all such that for all , is left invertible (i.e. there exists such that ).
-
4.
The largest ideal such that for all , is a unit in .
-
5.
(1) - (3) with “left” replaced by “right” and replaced by .
If is commutative and finitely generated
, then
The Jacobson radical can also be defined for non-unital rings.
To do this, we first define a binary operation on the ring
by for all .
Then is a monoid,
and the Jacobson radical is defined to be the largest ideal of
such that is a group.
If is unital, this is equivalent
to the definitions given earlier.
Title | Jacobson radical |
Canonical name | JacobsonRadical |
Date of creation | 2013-03-22 12:36:11 |
Last modified on | 2013-03-22 12:36:11 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 19 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 16N20 |
Related topic | Annihilator |
Related topic | RadicalOfAnIdeal |
Related topic | SimpleModule |
Related topic | Nilradical |
Related topic | RadicalTheory |
Related topic | QuasiRegularity |