- Artinian ring
There are two classes of rings that have very similar properties:
Emil Artinfirst 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:
* A ring is left Artinian if it satisfies the descending chain condition on left ideals.
* A ring is right Artinian if it satisfies the descending chain condition on right ideals.
* A ring is Artinian or two-sided Artinian if it is both left and right Artinian.
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.
Artin-Wedderburn theoremcharacterizes all simple rings that are Artinian: they are the matrix rings over a division ring. This implies that for simple rings, both left and right Artinian coincide.
Akizuki-Hopkins-Levitzki theorem, a left (right) Artinian ring is automatically a left (right) Noetherian ring.
* Charles Hopkins. Rings with minimal condition for left ideals. Ann. of Math. (2) 40, (1939). 712--730.
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