Co-coatomically supplemented modules

Abstract

It is shown that if a submodule N of M is co-coatomically supplemented and M/N has no maximal submodule, then M is a co-coatomically supplemented module. If a module M is co-coatomically supplemented, then every finitely M-generated module is a co-coatomically supplemented module. Every left R-module is co-coatomically supplemented if and only if the ring R is left perfect. Over a discrete valuation ring, a module M is co-coatomically supplemented if and only if the basic submodule of M is coatomic. Over a nonlocal Dedekind domain, if the torsion part T(M) of a reduced module M has a weak supplement in M, then M is co-coatomically supplemented if and only if M/T (M) is divisible and TP (M) is bounded for each maximal ideal P. Over a nonlocal Dedekind domain, if a reduced module M is co-coatomically amply supplemented, then M/T (M) is divisible and TP (M) is bounded for each maximal ideal P. Conversely, if M/T (M) is divisible and TP (M) is bounded for each maximal ideal P, then M is a co-coatomically supplemented module.