Inclusion between submodules

Question

Let $M$ be a module, $K$, $L$ and $N$ be submodules of $M$ such that $M=L+N$ and $L \cap K=(0)$ it is clear that $L+K$ is a submodule of $L+N$. Is there an inclusion or intersection relation between $K$ and $N$. Thanks for any help.

Answer 1

Try always to see what will happen in our familiar examples: vector spaces!

Let $M = \mathbf{R}^2 = \mathbf{R}x + \mathbf{R}y$ where $x, y$ are standard basis of $\mathbf{R}^2$. Set $L = \mathbf{R}x$ and $N = \mathbf{R}y$. If $K = \mathbf{R}(x + y)$ then you can see $L \cap K = \{ 0 \}$ but $K \not\subseteq N$ and $K \cap N = \{ 0 \}$ (trivial intersection).

In conclusion, the relations between $K$ and $N$ that you expected do not hold in general.