### Abstract

All rings are commutative with identity, and all modules are unital. The purpose of this article is to investigate multiplication von Neumann regular modules. For this reason we introduce the concept of nilpotent submodules generalizing nilpotent ideals and then prove that a faithful multiplication module is von Neumann regular if and only if it has no nonzero nilpotent elements and its Krull dimension is zero. We also give a new characterization for the radical of a submodule of a multiplication module and show in particular that the radical of any submodule of a Noetherian multiplication module is a finite intersection of prime submodules.

Original language | English |
---|---|

Pages (from-to) | 4620-4642 |

Number of pages | 23 |

Journal | Communications in Algebra |

Volume | 36 |

Issue number | 12 |

DOIs | |

Publication status | Published - Dec 2008 |

### Fingerprint

### Keywords

- Idempotent submodule
- Multiplication module
- Nilpotent submodule
- Pure submodule
- Von Neumann regular module

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Idempotent and nilpotent submodules of multiplication modules.** / Ali, Majid M.

Research output: Contribution to journal › Article

*Communications in Algebra*, vol. 36, no. 12, pp. 4620-4642. https://doi.org/10.1080/00927870802186805

}

TY - JOUR

T1 - Idempotent and nilpotent submodules of multiplication modules

AU - Ali, Majid M.

PY - 2008/12

Y1 - 2008/12

N2 - All rings are commutative with identity, and all modules are unital. The purpose of this article is to investigate multiplication von Neumann regular modules. For this reason we introduce the concept of nilpotent submodules generalizing nilpotent ideals and then prove that a faithful multiplication module is von Neumann regular if and only if it has no nonzero nilpotent elements and its Krull dimension is zero. We also give a new characterization for the radical of a submodule of a multiplication module and show in particular that the radical of any submodule of a Noetherian multiplication module is a finite intersection of prime submodules.

AB - All rings are commutative with identity, and all modules are unital. The purpose of this article is to investigate multiplication von Neumann regular modules. For this reason we introduce the concept of nilpotent submodules generalizing nilpotent ideals and then prove that a faithful multiplication module is von Neumann regular if and only if it has no nonzero nilpotent elements and its Krull dimension is zero. We also give a new characterization for the radical of a submodule of a multiplication module and show in particular that the radical of any submodule of a Noetherian multiplication module is a finite intersection of prime submodules.

KW - Idempotent submodule

KW - Multiplication module

KW - Nilpotent submodule

KW - Pure submodule

KW - Von Neumann regular module

UR - http://www.scopus.com/inward/record.url?scp=65249088982&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=65249088982&partnerID=8YFLogxK

U2 - 10.1080/00927870802186805

DO - 10.1080/00927870802186805

M3 - Article

AN - SCOPUS:65249088982

VL - 36

SP - 4620

EP - 4642

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 12

ER -