# Short exact sequence 0→Z→A→R→00\to \mathbb Z\to A \to \mathbb R \to 0

Does every short exact sequence $$0→Z→A→R→00\to \mathbb Z\to A \to \mathbb R \to 0$$ split in the category of Abelian groups?

The calculation of $$Ext1(Q,Z)\text{Ext}^1(\mathbb{Q}, \mathbb{Z})$$ can be found in this MO answer; in terms of just its isomorphism type the conclusion is that it’s an uncountable-dimensional vector space over $$Q\mathbb{Q}$$, abstractly isomorphic to $$R\mathbb{R}$$. It can also be written as a quotient $$AQ/Q\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}$$ where $$AQ≅ˆZ⊗Q\mathbb{A}_{\mathbb{Q}} \cong \hat{\mathbb{Z}} \otimes \mathbb{Q}$$ is the finite rational adeles.