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

Does every short exact sequence 0ZAR0 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 AQˆZQ is the finite rational adeles.

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

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