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 finite-dimensional;
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 infinite-dimensional, we can choose a sequence $\{x_{i}\}_{i\geq 1}$ of linearly independent elements. Let $U_{n}$ be the subspace spanned by $x_{1},\ldots,x_{n}$ and $V_{n}$ the subspace spanned by $x_{n+1},x_{n+2},\dots$ . 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 $(0)=\mathfrak{m}_{1}\dots\mathfrak{m}_{n}$ where the $\mathfrak{m}_{i}$ 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\mathfrak{m}_{1}\supset\mathfrak{m}_{1}\mathfrak{m}_{2}\supset\dots% \supset\mathfrak{m}_{1}\dots\mathfrak{m}_{n}=0

Each factor $\mathfrak{m}_{1}\dots\mathfrak{m}_{i-1}/\mathfrak{m}_{1}\dots\mathfrak{m}_{i}$ is a vector space over the field $A/\mathfrak{m}_{i}$ . 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, 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