PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (510)
Corrections (10)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
Jacobson radical (Definition)

The Jacobson radical $ J(R)$ of a unital ring $ R$ is the intersection of the annihilators of simple left $ R$-modules.

The following are alternative characterizations of the Jacobson radical $ J(R)$:

  1. The intersection of all left primitive ideals.
  2. The intersection of all maximal left ideals.
  3. The set of all $ t \in R$ such that for all $ r \in R$, $ 1-rt$ is left invertible (i.e. there exists $ u$ such that $ u(1-rt)=1$).
  4. The largest ideal $ I$ such that for all $ v \in I$, $ 1-v$ is a unit in $ R$.
  5. (1) - (3) with “left” replaced by “right” and $ rt$ replaced by $ tr$.

If $ R$ is commutative and finitely generated, then

$\displaystyle J(R)=\{x \in R \mid x^n=0 \hbox{ for some } n \in \mathbb{N} \} = \operatorname{Nil}(R). $

The Jacobson radical can also be defined for non-unital rings. To do this, we first define a binary operation $ \circ$ on the ring $ R$ by $ x\circ y=x+y-xy$ for all $ x,y\in R$. Then $ (R,\circ)$ is a monoid, and the Jacobson radical is defined to be the largest ideal $ I$ of $ R$ such that $ (I,\circ)$ is a group. If $ R$ is unital, this is equivalent to the definitions given earlier.



"Jacobson radical" is owned by yark. [ full author list (2) | owner history (1) ]
(view preamble)

View style:

See Also: annihilator, radical of an ideal, simple module, nilradical, radical theory, quasi-regularity


Attachments:
proof of characterizations of the Jacobson radical (Proof) by rspuzio
properties of the Jacobson radical (Result) by yark
Log in to rate this entry.
(view current ratings)

Cross-references: definitions, equivalent, group, monoid, binary operation, rings, finitely generated, commutative, unit, ideal, left invertible, left ideals, primitive ideals, characterizations, annihilators, intersection, unital ring
There are 17 references to this entry.

This is version 16 of Jacobson radical, born on 2002-04-20, modified 2008-01-03.
Object id is 2856, canonical name is JacobsonRadical.
Accessed 9645 times total.

Classification:
AMS MSC: 16N20 (Associative rings and algebras :: Radicals and radical properties of rings :: Jacobson radical, quasimultiplication)
Pending Errata and Addenda
None.
[ View all 3 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)