If $M_1\cap K=M_2\cap K$ and $M_1+K=M_2+K$ then $M_1=M_2$?

Question

I have a ring $R$ with $K\le M$ and submodules $M_1,M_2$. If we have that: $$M_1\cap K=M_2\cap K \text{ and } M_1+K=M_2+K$$ can we conclude that $M_1=M_2$?

I don't think that this is true but it seems like if "the stuff lost by qutienting by $K$ is the same and the stuff that is not lost is the same" then we conclude that the whole thing is the same.

Answer 1

No, it isn't true. Take $R = k$ some field, then take $M = k^2$, $M_1 = \langle(0, 1)\rangle$, $M_2 = \langle(1, 1)\rangle$, $K = \langle(1, 0)\rangle$.

Then we have $M_1 \cap K = M_2 \cap K = 0$, $M_1 + K = M_2 + K = M$, but $M_1 \neq M_2$.

Answer 2

By a assumption $M_1⊆M_2$:
If $m∈M_2⟹m∈M_2+K=M_1+K⟹m=n+k; n∈M_1, k∈K$
So $k=m-n∈M_2∩K=M_1∩K$
So $m=n+k∈M_1$