Suppose I have an $R$-module $P$ and let $M, N\in P$ be submodules. Furthermore, suppose that both $M$ and $N$ are artinian and noetherian. Is it always true that the quotient $P/M\cap N$ is also artinian and noetherian?
I definitely think it is not always true, but I failed to come up with a counter-example.
Any non-Artinian, non-Noetherian module containing two distinct simple submodules would work, because then $P/(M\cap N)\cong P$.
This suggests $R=F$ for a field $F$, and choose $P$ to be an infinite dimensional $F$ vector space, and $M\neq N$ any two distinct 1-dimensional subspaces of $P$.