Artinian ring
|
|
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals.
There are two classes of rings that have very similar properties:
- Rings whose underlying set is finite.
- Rings that are finite-dimensional vector spaces over fields.
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:
- 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.
The Artin-Wedderburn theorem characterizes 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.
By the Hopkins-Levitzski theorem, a left (right) Artinian ring is automatically a left (right) Noetherian ring.