Generalizations of circle plus-supplemented modules

Abstract

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