Generalizations of ⊕ -supplemented modules

We introduce ⊕ -radical supplemented modules and strongly ⊕ -radical supplemented modules (briefly, srs⊕-modules) as proper generalizations of ⊕ -supplemented modules. We prove that (1) a semilocal ring R is left perfect if and only if every left R-module is an ⊕ -radical supplemented module; (2) a commutative ring R is an Artinian principal ideal ring if and only if every left R-module is an srs⊕-module; (3) over a local Dedekind domain, every ⊕ -radical supplemented module is an srs⊕-module. Moreover, we completely determine the structure of these modules over local Dedekind domains.