Sum of two submodules of a module, each of which is a direct summand, is again a direct summand?

Question

Let $N,P$ be submodules of a module $M$ such that $N,P$ are direct summands of $M$.

Is it true that $N+P$ is also a direct summand of $M$ ?

Answer 1

The answer is no, here is a counterexample: Consider the $\mathbb Z$-module $M = \mathbb Z^2$ with the two submodules $N = \mathbb Z\cdot (1,0)$, $P = \mathbb Z \cdot (1,2)$. Both are direct summands since the generating elements can each be completed to a basis with $(0,1)$. We have $M/(N+P) \cong \mathbb Z/2\mathbb Z$, so the quotient map doesn't split and therefore $N+P$ is not a direct summand.