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