Let R be a commutative ring with 1.

An R-module K has the 'S' property if K/T = K implies that the submodule T is trivial.

By Fitting's Lemma any Noetherian module has the 'S' property. There exist non-Noetherian modules with this property. For example the infinite product of Z_{2}xZ_{3}xZ_{5}x... running over all of the primes has the 'S' property, but is not Noetherian.

I am curious if there is a characterization of these kinds of modules out there.