Does every short exact sequence 0→Z→A→R→0 split in the category of Abelian groups?

**Answer**

The calculation of Ext1(Q,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, abstractly isomorphic to R. It can also be written as a quotient AQ/Q where AQ≅ˆZ⊗Q is the finite rational adeles.

**Attribution***Source : Link , Question Author : Richard P. , Answer Author : Qiaochu Yuan*