Let $M$ be a finitely generated Artinian $R$-module. Then there exists a decomposition $M = \oplus_{i=1}^kM_i$ such that each $M_i$ is an indecomposable submodule of $M$. It is known that $\text{rad}(M) = \oplus_{i=1}^k\text{rad}(M_i)$ and $\text{soc}(M) = \oplus_{i=1}^k\text{soc}(M_i)$.
I'm able to prove that $\text{top}(M) = \oplus_{i=1}^k\text{top}(M_i)$, where $\text{top}(M) := M/\text{rad}(M)$, however I'm wondering if this result has been established in the literature? This is for my MSc thesis, and I would rather not include a proof of a result that is well-known.
Thanks.