How $(\mathbb R,+)$ is a decomposible group?

Question

We know that $(\mathbb R\setminus \{0\},.)$ is decomposible as it can be expressed as internal direct product of $\mathbb R^{+}$ and $\{-1,1\}$. I have also found somewhere that $(\mathbb R,+)$ is decomposible. Can anyone explain how to show it? Any help will be appreciated.

Answer 1

We can see $\mathbb{R}$ as a vector space over $\mathbb{Q}$ (so using rationals as scalars). This has a base (using the axiom of choice, there is no explicit description for such a base!) $B = \{x_i : i \in I \}$, that we can split in to linearly independent subsets, say $B_1 = \{x_i: i \in I_1 \}$ and $B_2 = \{x_i: i \in I_2\}$. Then $(\mathbb{R}, +)$ is a direct sum of the span of $B_1$ and the span of $B_2$.