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Artinian ring

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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

Fill this in later

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

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Properties of commutative rings