Let $M$ be an $A$-module with finite length, i.e., $\ell_{A}(M)< \infty .$ Also $A$ is a Noetherian ring. Then I want to show that $$\ell_{A}(M)= \sum_{p \in \text{Ass}(M)} \ell_{A_p}(M_p).$$
I was trying to show this if $\text{Ass}(M) $ is a singleton. I couldn't do much. I need some help. Thank you.