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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.

Hussein Eid
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  • Every (proper, nonzero) direct summand of Z+Z is cyclic. This follows from the structure theorem of finitely generated abelian groups. – Mariano Suárez-Álvarez May 09 '23 at 15:02
  • another thing you can consider is that every summand of a free module is equivalently a projective module. However, the projective modules over $\mathbb{Z}$ are free ($\mathbb{Z}$ is a PID). So now essentially you are done by your observation about the GCD. – Felix May 09 '23 at 15:04
  • @Enkidu I don't get what is meant by the statement 'So now essentially you are done by your observation about the GCD' . – Hussein Eid May 09 '23 at 15:13
  • you know that all non-trivial summands are cyclic and you classified all cyclic summands, sounds like you classified all summands. – Felix May 09 '23 at 15:16
  • How does it follow from the structure theorem of finitely generated abelian groups that any nonzero proper summand of $\mathbb{Z}\oplus \mathbb{Z}$ is cyclic?!. May you clarify please?! @MarianoSuárez-Álvarez – Hussein Eid May 09 '23 at 17:51

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