H Ansari-Toroghy, F FarshadifarOn comultiplication modules. Korean Ann Math, 25 (2) (), pp. 5. H Ansari-Toroghy, F FarshadifarComultiplication. Key Words and Phrases: Multiplication modules, Comultiplication modules. 1. Introduction. Throughout this paper, R will denote a commutative ring with identity . PDF | Let R be a commutative ring with identity. A unital R-module M is a comultiplication module provided for each submodule N of M there exists an ideal A of.

Author: | Tojak Dozshura |

Country: | China |

Language: | English (Spanish) |

Genre: | Spiritual |

Published (Last): | 7 April 2006 |

Pages: | 189 |

PDF File Size: | 13.70 Mb |

ePub File Size: | 15.52 Mb |

ISBN: | 415-9-88491-658-3 |

Downloads: | 15068 |

Price: | Free* [*Free Regsitration Required] |

Uploader: | Taut |

### Al-Shaniafi , Smith : Comultiplication modules over commutative rings

Let K be a non-zero graded submodule of M. Hence I is a gr -small ideal of R. Prices are subject to change without notice.

Let R be a G-graded ring and M a graded R – module. Volume 15 Issue 1 Janpp. Thus I is a gr -large ideal of R. Note first that K: If M is a gr – comultiplication gr – prime R – modulethen M is a gr – simple module. Let R be a G -graded commutative ring and M a graded R -module.

BoxIrbidJordan Email Other articles by this author: Some properties of graded comultiplication modules. Let N be a gr -finitely generated gr -multiplication submodule of M. User Account Log in Register Help. A respectful treatment of one another is important to us. Volume 8 Issue 6 Decpp. Therefore M is a gr -comultiplication module. Proof Let K be a non-zero graded submodule of M. Let R be a G – graded ring and M a gr – comultiplication R – module.

R N and hence 0: If every gr – prime ideal of R is contained in a unique gr – maximal ideal of Rthen every gr – second submodule of M contains a unique gr – minimal submodule of M. Volume 13 Issue 1 Jan Let R be G – graded ring and M a gr – comultiplication R – module.

We refer to [9] and [10] for these basic properties and more information on graded rings and modules. Volume 1 Issue 4 Decpp. Let R be a G – graded ring and M a gr – faithful gr – comultiplication module with the property 0: Proof Let N be a gr -finitely generated gr -multiplication submodule of M.

Proof Let N be a gr -second submodule of M. See all formats and pricing Online. First, we recall some basic properties of graded rings and modules which will be used in the sequel. Let I be an ideal of R. Then the following hold: Since M is gr -uniform, 0: Thus by [ 8Lemma 3. Graded multiplication modules gr -multiplication modules over commutative graded ring have been studied by many authors extensively see [ 1 — 7 ].

Let J be a proper graded ideal of R. This completes the proof because the reverse inclusion is clear. By [ 1Theorem 3.

Suppose first that N is a gr -small submodule of M. So I is a gr modiles ideal of R. Therefore we would like to draw your attention to our House Rules. Volume 5 Issue 4 Decpp. Let G be a group with identity e. An ideal of a G -graded ring need not be G -graded. Volume 6 Issue 4 Decpp. About the article Received: Here we will study the class of comultiplicatioj comultiplication modules and coultiplication some further results which are dual to classical results on graded multiplication modules see Section 2.

Let R be a G – graded ringM a gr – comultiplication R – module and 0: Volume 2 Issue 5 Octpp.

## Open Mathematics

Since N is a gr -second submodule of Mby [ 8Comultiplivation 3. Let N be a gr -second submodule of M. Volume 12 Issue 12 Decpp. Conversely, let N be a graded submodule of M. A graded R -module M is said to be gr – Artinian if satisfies the descending chain condition for graded submodules.

Therefore R omdules gr -hollow. Then M is a gr – comultiplication module if and only if M is gr – strongly self-cogenerated. Volume 3 Issue 4 Decpp.