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

The calculation of $\text{Ext}^1(\mathbb{Q}, \mathbb{Z})$ 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 $\mathbb{Q}$, abstractly isomorphic to $\mathbb{R}$. It can also be written as a quotient $\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}$ where $\mathbb{A}_{\mathbb{Q}} \cong \hat{\mathbb{Z}} \otimes \mathbb{Q}$ is the finite rational adeles.