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As the title asks: Is it possible to refine any ascending/descending chain to a composition series for a module? I am trying to prove that any module has a finite composition series iff it is both artinian and noetherian. (Please do not provide answer). And I had a thought I was unsure about. For any arbitrary chain

$$A_1 \subset A_2 \subset \dots \subset A_k \subset A_{k+1} \subset ...$$

can this be refined to a composition series for the Module containing them?

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As far as I know, composition series are defined to be series of finite length. In that case, you couldn't refine an infinite chain into a composition series.

You can refine any two finite series to a common refinement, though. So in particular, if you postulate the existence of one (finite) composition series, you can prove that there aren't any infinite ascending chains or descending chains.

How? Suppose you have an infinite strictly ascending or descending chain. Pick out any subsequence of elements to form a finite chain longer than your composition series. Slap on the zero submodule onto one end and the whole module onto the other end.

So now you have your composition series and another finite series that's longer. By the Schreier refinement theorem, there should be a common refinement of the two, and it should be at least as long as both series: but a composition series by definition can't be refined any further, so you have arrived at a contradiction.

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