Radical formula and prime submodules

Let B be a submodule of an R-module M. The intersection of all prime submodules of M containing B is denoted by rad(B). For every positive integer it, a generalization of E(B) denoted by E-n(B) of M will be introduced. Moreover, < E(B)> subset of < E,(B)> subset of rad(B). In this paper we will study the equality < E-n(B)> = rad(B). It is proved that if R is an arithmetical ring of finite Krull dimension n, then < E-n(B)> = rad(B). (C) 2006 Elsevier Inc. All rights reserved.