Artinian Ring - Examples

Examples

  • An Artinian integral domain is a field.
  • A ring with finitely many, say left, ideals is a left Artinian. In particular, a finite ring (e.g., ) is left and right Artinian.
  • Let k be a field. Then is Artinian for every positive integer n.
  • If I is a nonzero ideal of a Dedekind domain A, then is a principal Artinian ring.
  • For each, the full matrix ring over a left Artinian (resp. left Noetherian) ring R is left Artinian (resp. left Noetherian).

The ring of integers is a Noetherian ring but is not Artinian.

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