Totally cofinitely weak Rad-supplemented modules

Abstract

Let R be a ring and M be a left R-module. M is called cofinitely weak Rad-supplemented if every cofinite submodule of M has a weak Rad-supplement in M. In this paper, we will define totally cofinitely weak Rad-supplemented modules. In general, the finite sum of totally cofinitely weak Rad-supplemented modules need not to be totally cofinitely weak Radsupplemented. However a module totally cofinitely weak Rad-supplemented if and only if it is the direct sum of a semisimple module and a totally cofinitely weak Rad-supplemented module. We will prove a module M is totally cofinitely weak Rad-supplemented if and only if M/K is totally cofinitely weak Rad-supplemented for a linearly compact submodule K of M. Similarly, a module M is totally cofinitely weak Rad-supplemented if and only if M/U is totally cofinitely weak Rad-supplemented for a uniserial submodule U of M. © 2012 Academic Publications, Ltd.