Jacobson radical: Difference between revisions
(Started the page) |
No edit summary |
||
| Line 1: | Line 1: | ||
{{curing-ideal-defining function}} | |||
==Definition for commutative rings== | ==Definition for commutative rings== | ||
| Line 14: | Line 15: | ||
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]]. | ||
Revision as of 07:43, 9 August 2007
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 the intersection of all its maximal ideals.
Definition for noncommutative rings
The noncommutative case was considered by Jacobson while proving the famous Jacobson density theorem (Fill this in later).
Particular cases
Trivial ideal
A commutative unital ring whose Jacobson radical is trivial is termed a semisimple ring.