A variation of supplemented modules

Over a general ring, an R-module is w-supplemented if and only if amply w-supplemented. It is proved that over a local Dedekind domain, all modules are w-supplemented and over a non-local Dedekind domain, an R-module M is w-supplemented if and only if Soc(M)ll M or M=S_0 oplus (bigoplus_{i in I} K), where S_0 is a torsion, semisimple submodule of M and K is the field of quotients of R.