### Abstract

All rings are commutative with identity and all modules are unital. Let R be a ring and M an R-module. In our recent work [6] we investigated faithful multiplication modules and the properties they have in common with projective modules. In this article, we continue our study and investigate faithful multiplication and locally cyclic projective modules and give several properties for them. If M is either faithful multiplication or locally cyclic projective then M is locally either zero or isomorphic to R. We show that, if M is a faithful multiplication module or a locally cyclic projective module, then for every submodule N of M there exists a unique ideal Γ(N) ⊆ Tr(M) such that N = Γ(N)M. We use this result to show that the structure of submodules of a faithful multplication or locally cyclic projective module and their traces are closely related. We also use the trace of locally cyclic projective modules to study their endomorphisms.

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

Pages (from-to) | 195-214 |

Number of pages | 20 |

Journal | Communications in Algebra |

Volume | 41 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2013 |

### Fingerprint

### Keywords

- Locally cyclic projective module
- Multiplication module
- Ring of endormophisms
- Trace of a module

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Some Remarks on Multiplication and Projective Modules II.** / Ali, Majid M.

Research output: Contribution to journal › Article

*Communications in Algebra*, vol. 41, no. 1, pp. 195-214. https://doi.org/10.1080/00927872.2011.628724

}

TY - JOUR

T1 - Some Remarks on Multiplication and Projective Modules II

AU - Ali, Majid M.

PY - 2013/1

Y1 - 2013/1

N2 - All rings are commutative with identity and all modules are unital. Let R be a ring and M an R-module. In our recent work [6] we investigated faithful multiplication modules and the properties they have in common with projective modules. In this article, we continue our study and investigate faithful multiplication and locally cyclic projective modules and give several properties for them. If M is either faithful multiplication or locally cyclic projective then M is locally either zero or isomorphic to R. We show that, if M is a faithful multiplication module or a locally cyclic projective module, then for every submodule N of M there exists a unique ideal Γ(N) ⊆ Tr(M) such that N = Γ(N)M. We use this result to show that the structure of submodules of a faithful multplication or locally cyclic projective module and their traces are closely related. We also use the trace of locally cyclic projective modules to study their endomorphisms.

AB - All rings are commutative with identity and all modules are unital. Let R be a ring and M an R-module. In our recent work [6] we investigated faithful multiplication modules and the properties they have in common with projective modules. In this article, we continue our study and investigate faithful multiplication and locally cyclic projective modules and give several properties for them. If M is either faithful multiplication or locally cyclic projective then M is locally either zero or isomorphic to R. We show that, if M is a faithful multiplication module or a locally cyclic projective module, then for every submodule N of M there exists a unique ideal Γ(N) ⊆ Tr(M) such that N = Γ(N)M. We use this result to show that the structure of submodules of a faithful multplication or locally cyclic projective module and their traces are closely related. We also use the trace of locally cyclic projective modules to study their endomorphisms.

KW - Locally cyclic projective module

KW - Multiplication module

KW - Ring of endormophisms

KW - Trace of a module

UR - http://www.scopus.com/inward/record.url?scp=84872416859&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84872416859&partnerID=8YFLogxK

U2 - 10.1080/00927872.2011.628724

DO - 10.1080/00927872.2011.628724

M3 - Article

AN - SCOPUS:84872416859

VL - 41

SP - 195

EP - 214

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 1

ER -