Chiang Mai J. Sci. 2006; 33(3) 249 On Quasi-Primary Submodules Shahabaddin E. Atani * and Ahmad Y. Darani Department of Mathematics, University of Guilan, P.O. Box 1914 Rasht Iran *Author for Correspondence; e-mail : ebrahimi@guilan.ac.ir Chiang Mai J. Sci. 2006; 33(3) : 249 - 254 www.science.cmu.ac.th/journal-science/josci.html Contributed Paper Received: 4 January 2006 Accepted: 3 July 2006. ABSTRACT Let R be a commutative ring with non-zero identity. We define a proper submodule Nof an R -module M to be quasi-primary if P M N rad R = ) : ( is a prime ideal of R. In this case we also say that N is a P-quasi-primary submodule of M. A number of results concerning quasi-primary submodules are given. For example, we show that over a Prüfer domain of finite character R , every non-zero R-submodule of a module M is the intersection of finite number of quasi-primary submodules with incomparable radicals. Keywords and phrases: quasi-primary, multiplication, secondary. 1. I NTRODUCTION In this paper all rings are commutative rings with non-zero identity and all modules are unital. Quasi-primary ideals in a commutative ring have been introduced and studied by L. Fuchs in [8] (also see [9]). An ideal I of R is called quasi-primary if its radical (we will denote it by rad(I)) is a prime ideal. Here we study quasi-primary submodules of a module. The primary and quasi-primary submodules are different concepts. In fact, every primary submodule is quasi-primary, but a quasi-primary submodule need not be primary (see Example 2.2). Various properties of quasi-primary submodules of a module are considered. For example, in Theorem 2.10, we show that if N is a quasi-primary R- submodule of a representable module M, then M/N is secondary. In Theorem 3.4, we show that over a Prüfer domain of finite character R, if M is a finitely generated multiplication R-module, then every non-zero submodule of M is the product of finite number of pairwise comaximal quasi-primary submodules of M. We also prove, in Theorem 3.5, if is a faithful multiplication module over a commutative ring R, then every quasi- primary submodule of M contained in a prime submodule of M. Now we define the concepts that we will use. If R is a ring and N is a submodule of an R-module M, the ideal } : { N rM R r ⊆ ∈ will be denoted by ) : ( M N R . Then ) : 0 ( M R is the annihilator of M, ann(M). An R-module M is called a multiplication module if for each submodule N of M, N=IM for some ideal I of R. In this case we can take I = (N: R M). For an R-module M, we define the ideal ∑ ∈ = M m R M Rm M ) : ( ) ( θ . So if N is a submodule of M, then M M M ) ( θ = and N M N ) ( θ = (see [1]). A proper submodule N of M is primary (resp. prime) if for any R r ∈ and M m ∈ such that N rm ∈ , either N m ∈ or N M r n ⊆ for some n (resp. either N m ∈ or N rM ⊆ ). It is easy to show that if N is a primary submodule of M (resp. N is a prime submodule of M)