The splitting problem for coalgebras

Let R be a ring and let t be a torsion preradical, R is said to have the splitting property, provided that for every left R-module M, the torsion submodule t(M) of M is a direct summand of M. The characterization of rings with this property is a classical problem (in particular the Goldie and Dickson torsion theories have been studied) that for noncommutative rings remains open. We consider the problem for the algebra C*, associated to a coalgebra C, and the torsion preradical Rat. It is shown that if C* has the splitting property with respect a Rat, then C is finite dimensional.