Artinian ring

In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields.

A ring is left Artinian if it satisfies the descending chain condition on left ideals, right Artinian if it satisfies the descending chain condition on right ideals, and Artinian or two-sided Artinian if it is both left and right Artinian. For commutative rings and for the two classes of rings mentioned above these concepts conincide, but in general they are different.

The Artin–Wedderburn theorem characterizes all simple artinian rings as 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 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 it is Noetherian and if R / nil(R) is isomorphic to a direct product of finitely many fields, where nil(R) is the nilradical of R.^[1]

See also

• Artinian module
• Artinian ideal

Notes

1. ^ Atiyah & Macdonald 1969, Theorems 8.5 and 8.7

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