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?).