Short exact sequence of finitely generated R-modules

Question

Assume that M',M'' are finitely generated R-modules (here R is a commutative unitary ring) and M is another R-module. The initial problem is that if there exists a s.e.s. $0\rightarrow M'\xrightarrow{f} M\xrightarrow{g}M''\rightarrow 0 \text{ }(1)$ then we can deduce that M is also finitely generated and an answer to this can be found here Finitely generated modules in exact sequence. The answer was linear algebraish and I was wondering if we could somehow prove that the sequence spits so we get an isomorphism $M\cong M'\dot{\oplus} M''$ which would also imply that M IS Finitely generated. So the question is: Can we somehow prove that (1) splits assuming that M',M'' are finitely generated and if not is there a counterexample (this could happen if for example M'' is projective).