In this paper we characterize Q-modules and almost Q-modules. Next we estblish some equivalent conditions for an almost Q-module to be a Q-module. Using these results, some characterizations are given for Noetherian Q-modules. | Turk J Math 33 (2009) , 215 – 225. ¨ ITAK ˙ c TUB doi: Q-modules ¨ C. Jayaram and Unsal Tekir Abstract In this paper we characterize Q-modules and almost Q-modules. Next we estblish some equivalent conditions for an almost Q-module to be a Q-module. Using these results, some characterizations are given for Noetherian Q-modules. Key Words: Q-module, almost Q-module, Q-ring, almost Q-ring, Laskerian ring, Laskerian module, Noetherian spectrum, multiplication ideal and quasi-principal ideal. 1. Introduction Throughout this paper R denotes a commutative ring with identity and all modules are unital R -modules. L(R) denotes the lattice of all ideals of R . Throughout this paper M denotes a unital R -module. In this paper we introduce and study the concepts of Q -modules and almost Q -modules which are generalizations of Q -rings [4] and almost Q -rings [14]. We prove that a faithful R -module M is a Q -module if and only if R is a Q -ring and M is a multiplication module (see Theorem 1). It is shown that a faithful R -module M is a Q -module if and only if M is a Laskerian multiplication module in which every non maximal prime submodule is a finitely generated multiplication submodule (see Theorem 2). Next we establish several characterizations for almost Q -modules (see Theorem 3, Theorem 4, Theorem 5, Theorem 6 and Theorem 7). Using these results, some equivalent conditions are established for an almost Q -module to be a Q -module (see Theorem 8). Finally Noetherian Q -modules are characterized (see Theorem 9). 2. Basic notions √ For any x ∈ R , the principal ideal generated by x is denoted by (x). For any ideal I of R , I denotes the radical of I . Recall that an ideal I of R is called a multiplication ideal if for every ideal J ⊆ I , there exists an ideal K with J = KI . Multiplication ideals have been extensively studied; for example, see [1], [2] and [11]. If I is a multiplication ideal, then I is locally principal [1, Theorem 1 and .
