Let $M_R$ be any right $R$-module ($R$ is a ring with unity). An internal direct sum $\bigoplus_{i\in I}A_i$ of submodules of $M$ is called a local summand of $M$ if for every finite subset $F\subset I$, $\bigoplus_{i\in F}A_i$ is a summand of $M$.
Consider the right $\mathbb{Z}$-module $\mathbb{Z} \oplus \mathbb{Z}$. What are the local summands of this module ?!.
I have proved that:
A cyclic submodule $(a,b)\mathbb{Z}$ is a summand of $\mathbb{Z} \oplus \mathbb{Z}$ iff $\gcd(a,b)=1$.
Can this help here ?!.
Another question is: Is any direct summand of $\mathbb{Z}\oplus \mathbb{Z}$ cyclic ?!.
I appreciate any help. Thanks in advance.