| Artinian Ring |
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There are two classes of rings that have very similar properties:
Emil Artin first discovered that the descending chain condition for ideals generalizes both classes of rings simultaneously. Artinian rings are named after him. For noncommutative rings, we need to distinguish three very similar concepts:
For commutative rings, these concepts all coincide. They also coincide for the two classes of rings mentioned above, but in general they are different. There are rings that are left Artinian and not right Artinian, and vice versa. The s over a Division Ring . This implies that for simple rings, both left and right Artinian coincide. By the Akizuki-Hopkins-Levitzski Theorem , a left (right) Artinian ring is automatically a left (right) Noetherian Ring . |
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