As @Jeremy Rickard has mentioned, the impossibility was shown by Osofsky in 1968. But it is interesting that there is another article dating back to 1968 (again!) that proves the same thing. See Cateforis, Sandomerski, The singular submodule splits off, J. Algebra, 10 (1968), 149-165, Theorem 4.1. This article was received by J. Algebra on November 20, 1967 while Osofsky's paper was received by Proc. Amer. Math. Soc. on July 10, 1967. I hope this information meets the expectations.
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