Rad-\(\oplus\)-supplemented modules and cofinitely Rad-\(\oplus\)-supplemented modules. (Q2859250)

A module is supplemented (resp. \(\oplus\)-supplemented, Rad-supplemented) if every submodule of \(M\) has a supplement (resp. a supplement which is a direct summand, Rad-supplement) in \(M\). The authors define a (cofinitely) Rad-\(\oplus\)-supplemented module as one for which every (cofinite) submodule has a Rad-supplement which is a direct summand of \(M\). Among other results, the authors show that: 1) if \(M\) is a coatomic cofinitely Rad-\(\oplus\)-supplemented module, then \(M\) is an irredundant sum of local direct summands; 2) classes of (cofinitely) Rad-\(\oplus\)-supplemented modules are closed under finite direct sums.