RADICAL FORMULA AND WEAKLY PRIME SUBMODULES

Let B be a submodule of an R-module M. The intersection of all prime (resp. weakly prime) submodules of M containing B is denoted by rad(B) (resp. wrad(B)). A generalisation of (E(B)) denoted by U E(B) of M will be introduced. The inclusions < E(B)> subset of U E(B) subset of wrad(B) subset of rad(B) are motivations for studying the equalities U E(B) = wrad(B) and U E(B) = rad(B) in this paper. It is proved that if R is an arithmetical ring, then U E(B) = wrad(B). In Theorem 2.5, a generalisation of the main result of [11] is given.