Dergiler / Turkish Journal of Mathematics / 2004 / Cilt: 28 - Sayı: 3
Radical Submodules and Uniform Dimension of Modules
- Sayfa
- 255–270
- DOI
- —
Özet
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.
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.