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.

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