tailieunhanh - Radical submodules and uniform dimension of modules

We investigate the relations between a radical submodule N of a module M being a finite intersection of prime submodules of M and the factor module M/N having finite uniform dimension. It is proved that if N is a radical submodule of a module M over a ring R such that M/N has finite uniform dimension, then N is a finite intersection of prime submodules. | Turk J Math 28 (2004) , 255 – 270. ¨ ITAK ˙ c TUB Radical Submodules and Uniform Dimension of Modules P. F. Smith Abstract We investigate the relations between a radical submodule N of a module M being a finite intersection of prime submodules of M and the factor module M/N having finite uniform dimension. It is proved that if N is a radical submodule of a module M over a ring R such that M/N has finite uniform dimension, then N is a finite intersection of prime submodules. The converse is false in general but is true if the ring R is fully left bounded left Goldie and the module M is finitely generated. It is further proved that, in general, if a submodule N of a module M is a finite intersection of prime submodules, then the module M/N can have an infinite number of minimal prime submodules. 1. Introduction Throughout this note all rings are associative with identity and all modules are unital left modules. Let R be a ring and let M be an R-module. A submodule K of M is called prime if K 6= M and whenever r ∈ R and L is a submodule of M such that rL ⊆ K then rM ⊆ K or L ⊆ K. In this case, the ideal P = {r ∈ R : rM ⊆ K} is a prime ideal of R and we call K a P -prime submodule of M . For more information about prime submodules of M see, for example, [3]–[8] and [10]. A submodule N of a module M is called a radical submodule if N is an intersection of prime submodules of M . Note that radical submodules are proper submodules of M . Given a submodule N of a module M , a decomposition N = K1 ∩ · · · ∩ Kn in terms of submodules Ki (1 ≤ i ≤ n) of M , where n is a positive integer, is called irredundant 255 SMITH if N 6= K1 ∩ · · · ∩ Ki−1 ∩ Ki+1 ∩ · · · ∩ Kn for all 1 ≤ i ≤ n. In [11], a submodule N of a module M is said to have a prime decomposition if N is the intersection of a finite collection of prime submodules of M . Let N be a submodule of an R–module M such that N has a prime decomposition. Then N will be said to have a normal prime decomposition if .