chain conditions in vector spaces
From the theorem in the parent article  that an $A$module $M$ has a composition series^{} 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 space^{}. Then the following are equivalent^{}:

1.
$V$ is finitedimensional;

2.
$V$ has a composition series;

3.
$V$ satisfies the ascending chain condition^{} (acc);

4.
$V$ satisfies the descending chain condition^{} (dcc).
Proof.
Clearly (1) $\Rightarrow $ (2), since submodules^{} are just subspaces^{}. (2) $\Rightarrow $ (3) and (2) $\Rightarrow $ (4) from the parent article. So it remains to see that (3) $\Rightarrow $ (1) and (4) $\Rightarrow $ (1). But if $V$ is infinitedimensional, we can choose a sequence^{} ${\{{x}_{i}\}}_{i\ge 1}$ of linearly independent^{} elements. Let ${U}_{n}$ be the subspace spanned by ${x}_{1},\mathrm{\dots},{x}_{n}$ and ${V}_{n}$ the subspace spanned by ${x}_{n+1},{x}_{n+2},\mathrm{\dots}$. Then the ${U}_{i}$ form a strictly ascending infinite^{} family of subspaces, so $V$ does not satisfy the ascending chain condition; the ${V}_{i}$ 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 $\mathrm{(}\mathrm{0}\mathrm{)}\mathrm{=}{\mathrm{m}}_{\mathrm{1}}\mathit{}\mathrm{\dots}\mathit{}{\mathrm{m}}_{n}$ where the ${\mathrm{m}}_{i}$ are (not necessarily distinct) maximal ideals^{}. Then $A$ is Noetherian^{} if and only if $A$ is Artinian^{}.
Proof.
We have the sequence of ideals
$$A\supset {\U0001d52a}_{1}\supset {\U0001d52a}_{1}{\U0001d52a}_{2}\supset \mathrm{\dots}\supset {\U0001d52a}_{1}\mathrm{\dots}{\U0001d52a}_{n}=0$$ 
Each factor ${\U0001d52a}_{1}\mathrm{\dots}{\U0001d52a}_{i1}/{\U0001d52a}_{1}\mathrm{\dots}{\U0001d52a}_{i}$ is a vector space over the field $A/{\U0001d52a}_{i}$. 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 sequence^{}, 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. ∎
References
 1 M.F. Atiyah, I.G. MacDonald, Introduction to Commutative Algebra, AddisonWesley 1969.
Title  chain conditions in vector spaces 

Canonical name  ChainConditionsInVectorSpaces 
Date of creation  20130322 19:11:55 
Last modified on  20130322 19:11:55 
Owner  rm50 (10146) 
Last modified by  rm50 (10146) 
Numerical id  4 
Author  rm50 (10146) 
Entry type  Theorem 
Classification  msc 16D10 