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 ${\{x_i\}}_{i\ge 1}$ 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

$$A\supset {𝔪}_1\supset {𝔪}_1{𝔪}_2\supset \mathrm{\dots }\supset {𝔪}_1\mathrm{\dots }{𝔪}_n=0$$

Each factor ${𝔪}_1\mathrm{\dots }{𝔪}_{i-1}/{𝔪}_1\mathrm{\dots }{𝔪}_i$ is a vector space over the field $A/{𝔪}_i$. 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.∎