$\hspace{20pt}$Duplicate on overflow.
This question, in a way, extends this one. The question is what are some sufficient conditions on the abelian group $B$ so that if $B\cong A\oplus C$ and $f:A\rightarrow B$ and $g:B\rightarrow C$ are such that the sequence $$0\longrightarrow A\xrightarrow{\;\;f\;\;} B\xrightarrow{\;\;g\;\;} C\longrightarrow 0$$ is exact, then this sequence splits?
One of the answers to the linked question claims that a sufficient condition is that $B$ is finitely generated. Where can I find some more extensive, or ongoing work on similar results with weaker sufficient conditions on $B$? Also, if you know some other sufficient condition, please tell it in an answer.
Example:
Is $B$ being of finite rank (or at least $B\leq\mathbb{Q}^n$ for some $n\in\mathbb{N}$) a sufficient condition?
