Artinian ring: Difference between revisions
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Revision as of 02:21, 9 January 2007
Definition for commutative rings
Symbol-free definition
A commutative unital ring (or more generally a commutative ring) is termed Artinian if it satisfies the descending chain condition on ideals, that is, any descending chain of ideals stabilizes after a finite length.
Definition with symbols
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Definition for non-commutative rings
There are two notions:
- Left Artinian ring must satisfy the descending chain condition on left ideals
- Right Artinian ring must satisfy the descending chain condition on right ideals