Equivalence of two definitions of module length

Question

Usually, we say a module is of finite length if it has a composition series. We then use the composition series to define its length (after showing the Jordan Holder theorem).

I'm trying to work with an alternative definition and show that they are equal. For the sake of clarity, let's call this quantity width of a module (just to distinguish the two).

Def (proposed): The width of a module is the supremum of the length of all possible chains of submodules of M (by chains, I dont need successive quotients to be irreducible, unlike in composition series). A module is of finite width if this quantity is finite.

Now, I have shown the following results (starting from my definition)

• Let M be of finite width k, then any submodule has width at most k, with equality iff the submodule is M
• Let M be of finite width k, then any chain of width k must be a composition series. (ie finite width => finite length) Also, any composition series must have width k.
• Let M be of finite width k, then the Jorden Holder theorem holds.
• Let M be of finite width k, then any chain can be extended to a composition series by "slotting".

But I can't show this: If M has a composition series, then it has finite width. (ie, I cant show that finite length => finite width). All the results above assumes "finite width".

Attempt: I wanted to use finite length => Noetherian (then hopefully maybe implies finite width), but proofs of this that I found already loosely assumes finite length/existence of composition series of length k means no chain can be longer than k. In fact, I find most online sources to be treating these two conditions equivalently just like that - without prior justification.

This seems like it's clearly true, but I cant show it properly. Thanks!

Answer 1

(Quick note: Lattice theoretically, modules of finite composition $n$ have a chain with $n$ elements which is maximal. In the literature, we have already 'dualized' chains to antichains and defined the width of a module to be the maximal size of an antichain. Your choice of terms is fine for the purposes of our discussion, but I thought you might be interested in this width, too.)

First of all, you must be implicitly assuming that the chains in your definition are strictly ascending, or else everything has infinite width.

Suppose your module has finite composition length $n$. Since this implies the module is Artinian (and Noetherian) it's clear that the module doesn't have infinite width. So, find a strictly increasing chain of submodules from $0$ to $M$ of maximal length $m$. As you mentioned, you said this chain could be refined into a composition series, which would have at least $m$ terms, and so $m\leq n$. But you can't have $m<n$, because that would mean that there are composition series longer than your 'maximal' chain of length $m$. Thus, $m=n$.

The other direction is not too hard, if you haven't already solved it. If there is a maximal (strictly ascending) chain of $m$ elements in $M$, then it's obvious by corresponence that each factor is simple. That is, if $M_i\subseteq M_{i+1}$ is a link in the chain, the existence of a module between the two would lengthen the chain by $1$, contradicting the claim of maximality of the width. Thus $M_{i+1}/M_i$ is simple, and the original chain is a composition series.