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On dimension of modules

In this paper we prove the lying over and going down theoremsfor modules. Finally, we apply the above theorems and prove some results on the dimension of a module and itssubmodule. | Turk J Math 31 (2007) , 95 – 109. ¨ ITAK ˙ c TUB On Dimension of Modules S. Karimzadeh, R. Nekooei Abstract In this paper we prove the lying over and going down theorems for modules. Finally, we apply the above theorems and prove some results on the dimension of a module and its submodule. Key Words: Prime Submodule, Multiplication module, Dimension of a module. Introduction Throughout this note, all rings are commutative with identity and all modules are unital. For R-modules M and M , we denote all R-module homomorphisms of M into M by HomR (M, M ). For any submodule N of an R-module M , we define (N : M ) = {r ∈ R : rM ⊆ N } and denote (O : M ) by AnnR (M ). A submodule P of M is called prime if P = M , and whenever r ∈ R, m ∈ M and rm ∈ P , then m ∈ P or r ∈ (P : M ) [see 8]. It is easy to show that, if P is a prime submodule of an R-module M , then (P : M ) is a prime ideal of R. The sets of all prime submodules and proper maximal submodules of M are respectively denoted by Spec(M ) and M ax(M ). Following [4], we denote the intersection of all prime submodules of an R-module M by radM (0) and the intersections of all proper maximal submodules by √ Rad(M ). The radicals of R and an ideal I of R are denoted by N (R) and I, respectively. 2000 Mathematics Subject Classification: 13C13, 13C99. 95 KARIMZADEH, NEKOOEI An R-module M is called a multiplication module if for any submodule N of M there exists an ideal I of R such that N = IM . It is easy to check that M is a multiplication module if and only if N = (N : M )M for every submodule N of M (See [7]). Let R be a principal ideal domain (PID) and m and n be positive integers. Let A = (aij ) ∈ Mm×n (R) and F be the free R-module R(n). We shall use the notation A for the submodule N of F generated by the rows of A, and the notation (r1 , . . . , rm )A, ri ∈ R, for an element of N . In this paper we shall first prove the lying-over and going-down theorem for modules, and then prove results on the .