Every module of finite uniform dimension is a direct sum of (finitely many) indecomposable

Question

Crossposted on StackExchange on July 28 (no answer so far).

Let $R$ be a (commutative or non-commutative, associative, unital) ring. It is well known that any artinian or noetherian $R$-module $M$ can be factored into a direct sum of (finitely many) indecomposable submodules. Some months ago, I happened to note that the same conclusion carries over to the case where $M$ has finite uniform dimension (by Corollary (6.7)(1) in Lam's Lectures on Modules and Rings, every artinian or noetherian module has finite uniform dimension).

Question. I haven't yet been able to find a reference for this generalization; and that's precisely what I'm hoping for.

It's obvious to me that the result is nothing new (the proof is rather straightforward from the properties of the uniform dimension). I'm just surprised that I can't see the result being mentioned in any of the standard books (maybe because it's commonly stated in a more general form?).

Answer 1

I think that the focus is usually on finitely length modules (those that have both ACC and DCC on submodules) because of the uniqueness part of the Krull-Schmidt theorem. In general, indecomposable decompositions are quite ill-behaved without additional hypotheses.

The fact that uniform modules are finite direct sums of indecomposables follows quickly from Proposition 6.4 in Lam's "Lectures on Modules and Rings". To see this, suppose $M$ is any module which is not a finite direct sum of indecomposables. In particular $M$ is not indecomposable so $M=A_1\oplus B_1$ for some $A_1,B_1\neq 0$.

Now, it cannot be the case that $A_1$ and $B_1$ are both finite direct sums of indecomposables, thus without loss of generality $B_1$ is not a finite direct sum of indecomposables. So we can write $B_1=A_2\oplus B_2$ with $A_2,B_2\neq 0$. Repeating this process, we get $A_1\oplus A_2\oplus\cdots$ is an infinite direct sum inside $M$.

By Proposition 6.4, this means that $M$ does not have finite uniform dimension. Also note that there are indecomposable modules with infinite uniform dimension, so Proposition 6.4 characterizes finite uniform dimension using a property that is strictly stronger than having a finite indecomposable decomposition.