I am looking for a reference which will prove that a noetherian $A$-module $M$ over a noetherian ring $A$ has a filtration $0 = M_0 \subset \cdots \subset M_n = M$ with $M_i / M_{i-1} \cong A / \mathfrak{p}_i$ for each $i$. I am particularly interested in how this is related to primary decomposition and associated primes.
From what I notice immediately, $A / \mathfrak{p}$ injects into $M$ if and only if $M$ has $\mathfrak{p}$ as an associated prime, so that we can pass to $M/(A/\mathfrak{p})$ and induct. My tenuous guess is that $[M] = \sum_{i = 1}^n n [A / \mathfrak{p}_i]$ in the Grothendieck group of $K_0( \text{noetherian-}A \text{-mod})$ of noetherian $A$-modules, where $\mathfrak{p}_i$ are the associated primes of $M$.
Is there somewhere where I can find more of what this perspective would look like?