Let $R$ be a (commutative or non-commutative, associative, unital) ring. It is very well known that any artinian or noetherian $R$-module $M$ is a direct sum of (finitely many) indecomposable submodules. Some months ago, I just happened to note that the result generalizes 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). However, I haven't yet been able to find a reference for this generalization; and that's precisely what I'm hoping for with this question.
It's obvious to me that the result is nothing new. But I'm surprised that I can't see it mentioned in any of the standard books (maybe because it's commonly stated in a more general form).