Artinian Ring - Commutative Artinian Rings

Commutative Artinian Rings

Let A be a commutative Noetherian ring with unity. Then the following are equivalent.

• A is Artinian.
• A is a finite product of commutative Artinian local rings.
• A / nil(A) is a semisimple ring, where nil(A) is the nilradical of A.
• A has dimension zero.
• is finite and discrete.
• is discrete.

Let k be a field and A finitely generated k-algebra. Then A is Artinian if and only if A is finitely generated as k-module.

An Artinian local ring is complete. A quotient and localization of an Artinian ring is Artinian.