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.[1] For commutative rings the left and right definitions coincide, but in general they are distinct from each other.

The Wedderburn–Artin theorem characterizes every simple Artinian ring as a ring of matrices over a division ring. This implies that a simple ring is left Artinian if and only if it is right Artinian.

The same definition and terminology can be applied to modules, with ideals replaced by submodules.

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 counterexamples

The following two are examples of non-Artinian rings.

Modules over Artinian rings

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.[4]

Commutative Artinian rings

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

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.

An Artinian local ring is complete. A quotient and localization of an Artinian ring is Artinian.

Simple Artinian ring

One version of the Wedderburn–Artin theorem states that a simple Artinian ring A is a matrix ring over a division ring. Indeed,[8] let I be a minimal (nonzero) right ideal of A, which exists since A is Artinian (and the rest of the proof does not use the fact that A is Artinian). 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:

It is surjective. If it is not injective, then, say, with nonzero . Then, by the minimality of I, we have: . It follows:

,

which contradicts the minimality of k. Hence, and thus .

See also

Citations

  1. ^ Brešar 2014, p. 73
  2. ^ Clark, Theorem 20.11
  3. ^ Cohn 2003, 5.2 Exercise 11
  4. ^ Bourbaki 2012, VIII, p. 7
  5. ^ Atiyah & Macdonald 1969, Theorems 8.7
  6. ^ Atiyah & Macdonald 1969, Theorems 8.5
  7. ^ Atiyah & Macdonald 1969, Ch. 8, Exercise 2
  8. ^ Milnor 1971, p. 144

References

  • 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.
  • Clark, Pete L. "Commutative Algebra" (PDF). Archived from the original (PDF) on 2010-12-14.
  • Milnor, John Willard (1971), Introduction to algebraic K-theory, Annals of Mathematics Studies, vol. 72, Princeton, NJ: Princeton University Press, MR 0349811, Zbl 0237.18005