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 coincide, 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.

Although the descending chain condition appears dual to the ascending chain condition, in rings it is in fact the stronger condition. Specifically, a consequence of the Akizuki–Hopkins–Levitzki theorem is that a left (right) Artinian ring is automatically a left (right) Noetherian ring. This is not true for general modules, that is, an Artinian module need not be a Noetherian module.

Examples

• Let k be a field. Then is artinian for every positive integer n.

Commutative Artinian rings

Let A be a commutative noetherian ring with unity. Then the following are equivalent.

• A is artinian.
• A / nil(A) is isomorphic to a direct product of finitely many fields, where nil(A) is the nilradical of A.^[1]
• A has dimension zero.
• is finite and discrete.
• is finite.

Let k be a field and A finitely generated k-algebra. Then A is Artinian if and only if A is finitely generated as A-module.

See also

• Artin algebra
• 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