Submodules finitely generated if sum and intersection are finitely generated?

Question

Let $M,N$ be submodules of an R-module $L$, where $R$ is a commutative ring with unity. I would like to prove that if $M+N$ and $M \cap N$ are finitely generated so are $M$ and $N$. Trying to combine the generating sets of the sum and intersection did not bring me any results. Any idea how to approach this problem?

Answer 1

• For $A$-modules and homomorphisms $0\to M'\stackrel{u}{\to}M\stackrel{v}{\to}M''\to 0$ is exact, if $M'$ and $M''$ are fintely generated then $M$ is finitely generated.

• There is exact sequence: $0 \to M_1 \cap M_2 \to M_1 \oplus M_2 \to M_1 + M_2 \to 0 $ .