Let A is a Noetherian ring and M is a finitely generated A-module.
Show that $l_A$(M) $< \infty$ if and only if M is a direct sum of p-coprimary submodules of M for p $\in Ass_A$(M).
I came across this problem when trying to learn about associated primes and primary decomposition and I was a bit stumped on how to prove it. I couldn't find any references to this statement in the lecture notes and books I looked in. I'd appreciate any help with the proof / resources that might help!