chain conditions in vector spaces

From the theorem in the parent article - that an A-module M has a composition seriesMathworldPlanetmathPlanetmathPlanetmath if and only if it satisfies both chain conditions - it is easy to see that

Theorem 1.

Let k be a field, V a k-vector spaceMathworldPlanetmath. Then the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.
  2. 2.

    V has a composition series;

  3. 3.

    V satisfies the ascending chain conditionMathworldPlanetmathPlanetmathPlanetmath (acc);

  4. 4.

    V satisfies the descending chain conditionMathworldPlanetmathPlanetmath (dcc).


Clearly (1) (2), since submodulesMathworldPlanetmath are just subspacesPlanetmathPlanetmath. (2) (3) and (2) (4) from the parent article. So it remains to see that (3) (1) and (4) (1). But if V is infinite-dimensional, we can choose a sequencePlanetmathPlanetmath {xi}i1 of linearly independentMathworldPlanetmath elements. Let Un be the subspace spanned by x1,,xn and Vn the subspace spanned by xn+1,xn+2,. Then the Ui form a strictly ascending infiniteMathworldPlanetmath family of subspaces, so V does not satisfy the ascending chain condition; the Vi form a strictly descending infinite family of subspaces, so V does not satisfy the descending chain condition. ∎

This easily implies the following:

Corollary 1.

Let A be a ring in which (0)=m1mn where the mi are (not necessarily distinct) maximal idealsMathworldPlanetmathPlanetmath. Then A is NoetherianPlanetmathPlanetmath if and only if A is ArtinianPlanetmathPlanetmath.


We have the sequence of ideals


Each factor 𝔪1𝔪i-1/𝔪1𝔪i is a vector space over the field A/𝔪i. By the above theorem, each quotient satisfies the acc if and only if it satisfies the dcc. But by repeatedly applying the fact that in a short exact sequenceMathworldPlanetmath, the middle term satisfies the acc (dcc) if and only if both ends do, we see that A satisfies the acc if and only if it satisfies the dcc. ∎


  • 1 M.F. Atiyah, I.G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley 1969.
Title chain conditions in vector spaces
Canonical name ChainConditionsInVectorSpaces
Date of creation 2013-03-22 19:11:55
Last modified on 2013-03-22 19:11:55
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 4
Author rm50 (10146)
Entry type Theorem
Classification msc 16D10