Notes on reduced, artinian and multiplication modules

Let M be a unitary module over a commutative ring R with identity. In this paper we consider the concepts of Artinian, semi-Artinian, reduced and multiplication modules . Also we call an R-module M radical, if it has no maximal submodule. By P(M) we denote the sum of the radical submodules of M and we show that P(M/(P(M)) = ۰.