ON PRIMARY COMPACTLY PACKED MODULES MOHAMMED S. EL-ATRASH * AND ARWA E. ASHOUR ** Abstract. A proper submodule N of a module M over a ring R is com- pactly packed if for each family {Pα} α∈λ of prime submodules of M with N ⊆  α∈λ Pα,N ⊆ P β for some β ∈ λ. A module M is called compactly packed if every proper submodule is compactly packed. This concept was introduced in [17]. In this paper, we generalize this concept to primary sub- modules and introduce the concept of primary compactly packed modules. We also generalize the Prime Avoidance Theorem for modules that was proved in [13] to the Primary Avoidance Theorem for modules. In addition, we study various properties of primary compactly packed modules. Contents 1. Introduction 1 2. Primary Compactly Packed Modules 2 3. Primary Avoidance Theorem for Modules. 6 4. Minimal Primary Submodules 8 5. Basic Properties of Primary Radical Submodules 9 References 11 1. Introduction Let M be a unitary R-module, where R is a commutative rings with identity. A proper submodule N of M is primary if rm ∈ N, for r ∈ R and m ∈ M implies that either m ∈ N, or r n M ⊆ N for some positive integer n. In [17], the concept of compactly packed modules was introduced. We generalize this concept to the concept of primary compactly packed modules. A proper sub- module N of M is primary compactly packed (pcp) if for each family {P α } α∈λ of primary submodules of M with N ⊆  α∈λ P α ,N ⊆ P β for some β ∈ λ. A module M is called pcp if every submodule is pcp. Key words and phrases. primary submodules, primary compactly packed modules, primary radical submodules . 1