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I am reading the book of Bland, Rings and their Modules (2011). I have a question on Proposition 3.2.7 on page 85.

Proposition 3.2.7. A short exact sequence $S: 0\longrightarrow M_1\stackrel{f}\longrightarrow M\stackrel{g}\longrightarrow M_2\longrightarrow 0$ splits if and only if one of the following three equivalent conditions holds.
(1) Im f is a direct summand of $M$
(2) Ker g is a direct summand of $M$
(3) $M\cong M_1\oplus M_2$

I strongly doubt the equivalence of (1) and (3) in this context. Using (1), one can prove that the given ses $S$ splits. I think the proof of Bland is correct. See also Dauns, Modules and Rings (1994) op page 11.

But can one prove that $S$ splits using (3)? I think, Rotman contradicts this, see Rotman, An Introduction to Homological Algebra, 2nd edition (2009). In example 2.29 on page 54, Rotman constructs an ses $0\longrightarrow A\longrightarrow A\oplus M\longrightarrow M\longrightarrow 0$ that does not split.

Is my doubt correct and can someone elaborate on this?

user26857
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Steenis
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    There is another counterexample, here in this forum: http://math.stackexchange.com/questions/1082283/example-of-a-non-splitting-exact-sequence-0-%e2%86%92-m-%e2%86%92-m-oplus-n-%e2%86%92-n-%e2%86%92-0 – Steenis Aug 08 '16 at 20:46

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