GENERALIZED SUPPLEMENTED MODULES

Abstract

Let $R$ be a ring and $M$ a right $R$-module. It is shown that: (1) $M$ is Artinian if and only if $M$ is a GAS-module and satisfies DCC on generalized supplement submodules and on small submodules; (2) if $M$ satisfies ACC on small submodules, then $M$ is a lifting module if and only if $M$ is a GAS-module and every generalized supplement submodule is a direct summand of $M$ if and only if $M$ satisfies $(P^{*})$; (3) $R$ is semilocal if and only if every cyclic module is a GWS-module.