Does the equation $\operatorname{Ass}M=\operatorname{Ass}E(M)$ hold for non-finitely generated modules $M$?
Here $E(M)$ is the injective envelope of $M$ and $\operatorname{Ass}$ denotes the set of associated primes.
In a theorem in Bruns and Herzog's book "Cohen-Macaulay Rings" the equation has been proved for finitely generated modules but as much as I think I can't find where in the proof the assumption of being finitely generated has been used. I guess it must be true for all modules
Yes, it holds in general: If $R/{\mathfrak p}$ embeds into $M$, then also into $E(M)$ since $M\subset E(M)$. Conversely, if $R/{\mathfrak p}$ embeds into $E(M)$, then a nonzero submodule $L\subset R/{\mathfrak p}$ embeds into $M$ since $M\subset E(M)$ is essential. Assuming wlog that $L$ is cyclic, we have $L\cong R/{\mathfrak p}$ since ${\mathfrak p}$ is the annihilator of any non-zero element of $R/{\mathfrak p}$. Hence $R/{\mathfrak p}\cong L$ embeds into $M$.