Split Short Exact Sequences - Bland - Proposition 3.2.7 - Three equivalent condiitions

[Euge]

That does not make sense, since any counterexample to $(3)\implies (1)$ is a a counterexample to the implication that $(3)$ implies the ses splits; for a split exact sequence necessarily satisfies $(1), (2)$, and $(3)$. Seeing that you want to stop the discussion, I'll leave it to other users address your concerns.

 

[Math Amateur]

That does not make sense, since any counterexample to $(3)\implies (1)$ is a a counterexample to the implication that $(3)$ implies the ses splits; for a split exact sequence necessarily satisfies $(1), (2)$, and $(3)$. Seeing that you want to stop the discussion, I'll leave it to other users address your concerns.

Thanks to Euge and Steenis for clarifying matters ...

There are some further helpful and informative posts on the Physics Forums in the sub-forum Linear and Abstract Algebra ... here

https://www.physicsforums.com/threa...-bland-proposition-3-2-7.881174/#post-5538854

Hope that helps ...

Peter