In this paper we introduce the concept of primary finitely compactly packed modules, which generalizes the concept of primary compactly packed modules. We first find the conditions that make the primary finitely compactly packed modules primary compactly packed. | Turk J Math 32 (2008) , 315 – 324. ¨ ITAK ˙ c TUB Primary Finitely Compactly Packed Modules and S-Avoidance Theorem for Modules Arwa Eid Ashour Abstract In this paper we introduce the concept of primary finitely compactly packed modules, which generalizes the concept of primary compactly packed modules. We first find the conditions that make the primary finitely compactly packed modules primary compactly packed. Also, several results on the primary finitely compactly packed modules are proved. In addition, the necessary and sufficient conditions for an R−module M to be primary finitely compactly packed are investigated. Finally, we introduce the S-Avoidance Theorem for modules. Key Words: Primary submodules, primary compactly packed modules, primary finitely compactly packed modules, s-prime submodules, S-Avoidance Theorem for modules. 1. Introduction Let M be a unitary R−module, where Ris a commutative ring with identity. A proper submodule N of M is primary if rm ∈ N for r ∈ Rand m ∈ M implies that either m ∈ N or r n M ⊆ N for some positive integer n. It is known that a proper submodule N of an R−module M is primary compactly packed (pcp) if for each family {Pα }α∈λ of primary submodules of M with N ⊆ ∪ Pα , ∃β ∈ λ such that N ⊆ Pβ . A moduleM is called pcp if α∈λ every proper submodule of M is pcp; see [3]. We generalizes the concept of pcp modules to the concept of primary finitely compactly packed (pfcp) modules. Thus we say that a proper submodule N of an R−module M is pfcp if for each family {Pα }α∈λ of primary n submodules ofM with N ⊆ ∪ Pα , ∃α1 , α2, ., αn ∈ λ such that N ⊆ ∪ Pαi . A module α∈λ i=1 315 ASHOUR M is said to be pfcp if every proper submodule of M is pfcp. In section 2 of this paper, we give some examples of pfcp modules and find the relation between pcp modules and pfcp modules. We also find the conditions that make a pfcp module pcp. In section 3, we investigate some properties of pfcp modules. We also find the necessary and sufficient .
