- Artinian ring
In

abstract algebra , an**Artinian ring**is a ring that satisfies thedescending chain condition on ideals. They are also called**Artin rings**.There are two classes of rings that have very similar properties:

* Rings whose

underlying set s are finite.

* Rings that are finite-dimensionalvector space s 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 allsimple ring s that are Artinian: they are thematrix ring s over adivision ring . This implies that for simple rings, both left and right Artinian coincide.By the

Akizuki-Hopkins-Levitzki theorem , a left (right) Artinian ring is automatically a left (right)Noetherian ring .**ee also***

Artinian module **References*** Charles Hopkins. Rings with minimal condition for left ideals. Ann. of Math. (2) 40, (1939). 712--730.

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