On Primal Compactly Packed Modules

Abstract

Let R be a commutative ring with identity and let M be a unitary R-module. A proper submodule N of M is compactly packed if for each family {Pα}α∈Δ of prime submodules of M with N ⊆ Uα∈Δ Pα, N ⊆ Pβ for some β ∈ Δ. A module M is called compactly packed if every proper submodule of M is compactly packed. This concept was introduced in 1998 by Al-Ani (see [1]). The concept of compactly packed modules was generalized in 2005 to The concept of primary compactly packed modules by Ashour. Also she introduced the Primary Avoidance Theorem for modules (see [5]). In this thesis we recall the concept of primal submodules which is a generalization of the concepts of primary submodules. And we generalize the concept of primary compactly packed modules to the concept of primal compactly packed modules. We also generalize the Primary Avoidance Theorem for modules that was proved in [5] to the Primal Avoidance Theorem for modules. In addition to proving several results concerning primal submodules over Boolean rings, S-closed subsets of modules and primary compactly packed Noetherian modules.