Showing an abelian group indecomposable

Question

I want to show that $\mathbb Q$ as an abelian group or as an $\mathbb Z$- module is indecomposable. An $R$-module $M$ is said to be indecomposable if it cannot be written as a direct sum of its non-trivial(not $0$ or $M$ itself) submodule. Any hint how to proceed?

Answer 1

A direct sum $M \oplus N$ always has a pair of submodules that intersect only at the identity, namely $\{ (m,0) : m \in M\}$ and $\{(0,n):n\in N\}$.

But any two nontrivial subgroups of $\mathbb Q$ have infinite intersection.

Answer 2

Theorem 3.1.5[Cohen Macaulay rings-Bruns and Herzog] Let $R$ be a ring and $I$ an $R$ module. If $R$ is a principal domain and $I$ is divisible, then $I$ is injective.

By theorem $\mathbb{Q}$ is an injective $\mathbb{Z}$ module

Theorem 3.2.6[Cohen Macaulay rings-Bruns and Herzog] Let $R$ be a Noethering ring.

$(a)$ For all $p \in spec(R)$ the module $E(R/p)$ is indecomposable (Where $E(R/p)$ is injective hull of $R/p$)

put $R=\mathbb{Z}$ and $M=\mathbb{Q}$ , since $0 \in spec(R)$. $E(R/<0>)=\mathbb{Q}=M$ ( Injective hull of integral domain is field of fractions see)