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: | Brajind Kashura |

Country: | Liberia |

Language: | English (Spanish) |

Genre: | Video |

Published (Last): | 10 August 2009 |

Pages: | 354 |

PDF File Size: | 2.89 Mb |

ePub File Size: | 11.37 Mb |

ISBN: | 235-4-33854-578-7 |

Downloads: | 10916 |

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

Uploader: | Shajora |

If M is a gr – comultiplication gr – prime R – modulethen M is a gr – simple module. Let N be a gr -finitely generated gr -multiplication submodule of M. By [ 8Theorem 3. About the article Received: Volume 11 Issue 12 Decpp.

Proof Suppose mdoules that N is a gr -small mmodules of M. Therefore we would like to draw your attention to our House Rules.

Let R be a G – graded ringM a gr – comultiplication R – module and 0: Volume 2 Issue 5 Octpp. Since N is a gr -second submodule of Mby [ 8Proposition 3.

Let R be a G -graded commutative ring and M a graded R -module. The following lemma is known see [12] and [6]but we write it here for the sake of references. Let R be a G-graded ring and M a graded R – module. Proof Let J be a proper graded ideal of R.

## There was a problem providing the content you requested

Volume 13 Issue 1 Jan By [ 1Theorem moddules. Proof Suppose first that N is a gr -large submodule of M.

comultiplidation Since N is a gr -small submodule of M0: As a dual concept of gr -multiplication modules, graded comultiplication modules gr -comultiplication modules were introduced and studied by Ansari-Toroghy and Farshadifar [8]. Then M is a gr – comultiplication module if and only if M is gr – strongly self-cogenerated.

Let G be a group with identity e. Volume 12 Issue modulles Decpp. Prices do not include postage and handling if applicable. Abstract Let G be a group with identity e. Prices are subject to change without notice. Volume 5 Issue 4 Decpp. Let N be a gr -second submodule of M.

This completes the proof because the reverse inclusion modiles clear. Note first that K: Let G be a group with identity e and R be a commutative ring with identity 1 R. Volume 6 Issue 4 Decpp. Suppose first that N is a gr -small submodule of M. Let I be an ideal of R.

### Some properties of graded comultiplication modules : Open Mathematics

Therefore R is gr -hollow. Proof Let N be a gr -second submodule of M. A respectful treatment of one another is important to us. Therefore M is gr -uniform. A similar argument yields a similar contradiction and thus completes the proof. Suppose first that M is gr -comultiplication R -module and N a graded submodule of M.

By[ 8Lemma 3. Then M is gr – hollow module. Then M is gr – uniform if and only if R is gr – hollow. A graded R -module M is said to be gr – simple if 0 and M are its only graded submodules. Some properties of graded comultiplication modules.

## Open Mathematics

First, we recall some basic properties of graded rings and modules which will be used in the sequel. By using the comment function on degruyter. Let K be a non-zero graded submodule of M. Since M is gr -uniform, 0: Hence I is a gr -small ideal of R. Let R be a G -graded ring and M a graded R -module. It follows that 0: Let R be a G – graded ring and M a graded R – module. Graded comultiplication module ; Graded multiplication module ; Graded submodule.

Volume 3 Issue 4 Decpp. Proof Note first that K: