Jacobson radical: Difference between revisions

Revision as of 22:38, 2 February 2008

This article defines an ideal-defining function, viz a rule that inputs a commutative unital ring and outputs an ideal of that ring

Definition for commutative rings

Symbol-free definition

The Jacobson radical of a commutative unital ring is defined in the following equivalent ways:

It is the intersection of all its maximal ideals
It is the set of those elements which are in the kernel of every homomorphism to a field

Related notions

Nilradical: The nilradical is defined as the intersection of all prime ideals, and also as the set of all nilpotent elements. The nilradical is contained in the Jacobson radical. The two are equal for Jacobson rings (and for some other rings as well) but are not equal in general.

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 rings, then the Jacobson radical of the subring equals the intersection of the subring with the Jacobson radical of the ring.

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 ===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]].