Do you know how to prove that for a ring $R$, and a bilateral ideal $I$ of $R$, if $M$ is a coherent left $R$-module, then $M/IM$ is a coherent $R/I$-module?
I know how to prove the fact if $I$ annihilates $M$:
In that case, $_RM$ is finitely generated implies $_{R/I}M$ is f.g. too.
Furthermore, for a $_{R/I}N$ submodule f.g. of $_{R/I}M$, we have $_RN$ is f.g., and then is a submodule f.g. of $_RM$ which is coherent. Then, $_RN$ is finitely presented. It follows that $_{R/I} (N\otimes_RR/I)$ is f.p.
But $_{R/I} (N\otimes_RR/I)\cong N/IN\cong N$, and the result follows.
I don’t know how to proceed if the hypothesis of $I$ annihilates $M$ is missing.
Could you, please, give me an advice?
Thanks!