Co-coatomically supplemented modules

Abstract

The purpose of this study to define co-coatomically supplemented modules, -cocoatomically supplemented modules, co-coatomically weak supplemented modules and co-coatomically amply supplemented modules and examine them over arbitrary rings and over commutative Noetherian rings, in particular over Dedekind domains. Motivated by cofinite submodule which is defined by R. Alizade, G. Bilhan and P. F. Smith, we define co-coatomic submodule. A proper submodule is called co-coatomic if the factor module by this submodule is coatomic. Then we define co-coatomically supplemented module. A module is called co-coatomically supplemented if every co-coatomic submodule has a supplement in this module. Over a discrete valuation ring, a module is co-coatomically supplemented if and only if the basic submodule of this module is coatomic. Over a non-local Dedekind domain, if a reduced module is co-coatomically amply supplemented then the factor module of this module by its torsion part is divisible and P-primary components of this module are bounded for each maximal ideal P. Conversely, over a non-local Dedekind domain, if the factor module of a reduced module by its torsion part is divisible and P-primary components of this module are bounded for each maximal ideal P, then this module is co-coatomically supplemented. A ring R is left perfect if and only if any direct sum of copies of the ring is -co-coatomically supplemented left R-module. Over a discrete valuation ring, co-coatomically weak supplemented and co-coatomically supplemented modules coincide. Over a Dedekind domain, if the torsion part of a module has a weak supplement in this module, then the module is co-coatomically weak supplemented if and only if the torsion part is co-coatomically weak supplemented and the factor module of the module by its torsion part is co-coatomically weak supplemented. Every left R-module is co-coatomically weak supplemented if and only if the ring R is left perfect.