Let $M=\Bbb Z^2$ . If $N$ and $P$ internal direct summands of $M$ , then does it follow that $N+P$ is also an internal direct summand of $M$

Question

Let $M=\Bbb Z^2$ . If $N$ and $P$ internal direct summands of $M$ , then does it follow that $N+P$ is also an internal direct summand of $M$.

My attempt :

$\Bbb Z^2=\Bbb Z \oplus \Bbb Z$. Then $N=P=\Bbb Z$. Then isn't $N+P=M$ ? Then it's not an internal direct summand of $M$.

Is that all ? Perhaps I have misunderstood the question. Please point out mistakes.

Answer 1

Both $N:=\big\langle (1,0)\big\rangle$ and $P:=\big\langle (1,2)\big\rangle$ are internal direct summands of $M=\mathbb{Z}^2$, with a complementary direct summand $\big\langle (0,1)\big\rangle$. However, $N+P=\mathbb{Z}\times (2\mathbb{Z})$ is not an internal direct summand of $M$.