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.
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