Jacobson radical - Revision history

Vipul: 3 revisions

2008-05-12T16:24:16Z

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Vipul at 22:38, 2 February 2008

2008-02-02T22:38:55Z

← Older revision		Revision as of 22:38, 2 February 2008
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	===Symbol-free definition===		===Symbol-free definition===

	The '''Jacobson radical''' of a [[commutative unital ring]] is the ~~intersection of all its [[maximal ideal]]s.~~		The '''Jacobson radical''' of a [[commutative unital ring]] is defined in the following equivalent ways:

	~~==Definition for noncommutative rings==~~		* It is the intersection of all its [[maximal ideal]]s
			* It is the set of those elements which are in the kernel of every homomorphism to a field

	~~The noncommutative case was considered by Jacobson while proving the famous [[Jacobson density theorem]] ({{fillin}}).~~		==Related notions==

	~~==Particular cases==~~		* [[Nilradical]]: The nilradical is defined as the intersection of all [[prime ideal]]s, and also as the set of all [[nilpotent element]]s. The nilradical is contained in the Jacobson radical. The two are equal for [[Jacobson ring]]s (and for some other rings as well) but are not equal in general.

	~~===Trivial ideal===~~		A key difference between the Jacobson radical and the nilradical is that there is in general no way to relate the Jacobson radical of a ring with that of a subring; however, the nilradical of a subring is precisely the intersection of the subring with the nilradical of the whole ring (this is clear from the characterization of the nilradical as the set of nilpotent elements). Thus, if both the ring and the subring are [[Jacobson ring]]s, then the Jacobson radical of the subring equals the intersection of the subring with the Jacobson radical of the ring.

	A ~~commutative unital ring whose~~ Jacobson radical is ~~trivial~~ is ~~termed~~ a [[~~semisimple~~ ring]].

Vipul at 07:43, 9 August 2007

2007-08-09T07:43:13Z

← Older revision		Revision as of 07:43, 9 August 2007
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	==Definition for commutative rings==		==Definition for commutative rings==

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	A commutative unital ring whose Jacobson radical is trivial is termed a [[semisimple ring]].		A commutative unital ring whose Jacobson radical is trivial is termed a [[semisimple ring]].

	~~[[Category: Ideal-defining functions on commutative rings]]~~

Vipul: Started the page

2007-01-09T05:35:02Z

Started the page

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==Definition for commutative rings==

===Symbol-free definition===

The '''Jacobson radical''' of a [[commutative unital ring]] is the intersection of all its [[maximal ideal]]s.

==Definition for noncommutative rings==

The noncommutative case was considered by Jacobson while proving the famous [[Jacobson density theorem]] ({{fillin}}).

==Particular cases==

===Trivial ideal===

A commutative unital ring whose Jacobson radical is trivial is termed a [[semisimple ring]].

[[Category: Ideal-defining functions on commutative rings]]