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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) ]
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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
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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 9272 times total.

Classification:
AMS MSC: 16N20 (Associative rings and algebras :: Radicals and radical properties of rings :: Jacobson radical, quasimultiplication)
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