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Here, Martin Brandenburg says it is not true that "Every short exact sequence of the form $0 \to A \to A \oplus B \to B \to 0$ splits." Then Mohan says in comments that "As a positive result,

If $0 \to A \to A \oplus B \to B \to 0$ is an exact sequence of finitely generated modules over a commutative Noetherian ring, then the exact sequence does split."

Mohan adds as an answer to my comment "The proof, while not trivial, can be worked out and I would be happy to post one somewhere (how?) if you so desire."
Now Mohan and other friends can you please...
Thank you.

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    It seems that a reference is given in Neil Epstein's comment which is 3 comments above Mohan's first comment in the answer you refer to. (Which is this, for people's convenience: (Theorem 1 from T. Miyata, Note on direct summands of modules, J. Math. Kyoto Univ. 7 (1967) 65-69).) – Peter Samuelson Feb 12 '16 at 17:08
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    On the analogous question for finite groups: http://mathoverflow.net/questions/80002/ – YCor Feb 12 '16 at 17:26
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    See my answer here: http://mathoverflow.net/questions/167701/do-all-exact-sequences-0-rightarrow-a-rightarrow-a-oplus-b-rightarrow-b-ri/167706#167706 – Steven Landsburg Feb 12 '16 at 21:41

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