Artinian ring
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In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings.
There are two classes of rings that have very similar properties:
- Finite Rings.
- 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 Akizuki-Hopkins-Levitzki theorem, a left (right) Artinian ring is automatically a left (right) Noetherian ring.
Although the descending-chain condition appears similar to the ascending chain condition, in commutative rings it is in fact enormously stronger. Every Artinian commutative ring is automatically Noetherian; a direct characterization of Artinian rings is that a commutative ring R is Artinian if and only if R is Noetherian and R / nil(R) is isomorphic to a direct product of finitely many fields, where nil(R) is the nilradical of R.[1]
[edit] See also
[edit] Notes
[edit] References
- Charles Hopkins. Rings with minimal condition for left ideals. Ann. of Math. (2) 40, (1939). 712--730.
- Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8
