We will call a submodule $A$ a direct summand of $K$ if there exists a submodule $B$ such that $A \oplus B = K$. I think this is a question that can be formulated in terms of rank of a proper free sumbmodules but I am not sure how to ask it.
Consider the $\mathbb{Z}$-module $\mathbb{Z} \oplus \mathbb{Z}$. Is there an example of two submodules of $A,B$ of $\mathbb{Z} \oplus \mathbb{Z}$ such that $A$ and $B$ are direct summands of $\mathbb{Z} \oplus \mathbb{Z}$ but $A+B$ is not a direct summand of $\mathbb{Z} \oplus \mathbb{Z}$?
I first thought that $\mathbb{Z}\oplus 0$ and $ 0 \oplus \mathbb{Z}$ was an example until I realized every module is a direct summand of itself...