In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings 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. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition.
Precisely, 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 the left and right definitions coincide, but in general they are distinct from each other.
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 (resp. right) Artinian ring is automatically a left (resp. right) Noetherian ring. This is not true for general modules; that is, an Artinian module need not be a Noetherian module.
Examples and counterexamplesEdit
- An integral domain is Artinian if and only if it is a field.
- A ring with finitely many, say left, ideals is left Artinian. In particular, a finite ring (e.g., ) is left and right Artinian.
- Let k be a field. Then is Artinian for every positive integer n.
- Similarly, is an Artinian ring with maximal ideal .
- Let be an endomorphism between a finite-dimensional vector space V. Then the subalgebra generated by is a commutative Artinian ring.
- If I is a nonzero ideal of a Dedekind domain A, then is a principal Artinian ring.
- For each , the full matrix ring over a left Artinian (resp. left Noetherian) ring R is left Artinian (resp. left Noetherian).
The following two are examples of non-Artinian rings.
- If R is any ring, then the polynomial ring R[x] is not Artinian, since the ideal generated by is (properly) contained in the ideal generated by for all natural numbers n. In contrast, if R is Noetherian so is R[x] by the Hilbert basis theorem.
- The ring of integers is a Noetherian ring but is not Artinian.
Modules over Artinian ringsEdit
Let M be a left module over a left Artinian ring. Then the following are equivalent (Hopkins' theorem): (i) M is finitely generated, (ii) M has finite length (i.e., has composition series), (iii) M is Noetherian, (iv) M is Artinian.
Commutative Artinian ringsEdit
Let A be a commutative Noetherian ring with unity. Then the following are equivalent.
- A is Artinian.
- A is a finite product of commutative Artinian local rings.
- A / nil(A) is a semisimple ring, where nil(A) is the nilradical of A.
- Every finitely generated module over A has finite length. (see above)
- A has Krull dimension zero. (In particular, the nilradical is the Jacobson radical since prime ideals are maximal.)
- is finite and discrete.
- is discrete.
Let k be a field and A finitely generated k-algebra. Then A is Artinian if and only if A is finitely generated as k-module.
Simple Artinian ringEdit
A simple Artinian ring A is a matrix ring over a division ring. Indeed, let I be a minimal (nonzero) right ideal of A. Then, since is a two-sided ideal, since A is simple. Thus, we can choose so that . Assume k is minimal with respect that property. Consider the map of right A-modules:
which contradicts the minimality of k. Hence, and thus .
- Brešar 2014, p.73
- Theorem 20.11. of http://math.uga.edu/~pete/integral.pdf Archived 2010-12-14 at the Wayback Machine
- Cohn 2003, 5.2 Exercise 11
- Bourbaki, VIII, pg 7
- Atiyah & Macdonald 1969, Theorems 8.7
- Atiyah & Macdonald 1969, Theorems 8.5
- Atiyah & Macdonald 1969, Ch. 8, Exercise 2.
- Milnor, John Willard (1971), Introduction to algebraic K-theory, Annals of Mathematics Studies, vol. 72, Princeton, NJ: Princeton University Press, p. 144, MR 0349811, Zbl 0237.18005
- Auslander, Maurice; Reiten, Idun; Smalø, Sverre O. (1995), Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, doi:10.1017/CBO9780511623608, ISBN 978-0-521-41134-9, MR 1314422
- Bourbaki, Nicolas (2012). Algèbre. Chapitre 8, Modules et anneaux semi-simples. Heidelberg: Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-35315-7.
- 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
- Cohn, Paul Moritz (2003). Basic algebra: groups, rings, and fields. Springer. ISBN 978-1-85233-587-8.
- Brešar, Matej (2014). Introduction to Noncommutative Algebra. Springer. ISBN 978-3-319-08692-7.