### COMULTIPLICATION MODULES PDF

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.

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By using the comment function on degruyter. Since M is gr -uniform, 0: Note first that K: A graded R -module M is said to be gr – uniform resp. R N and hence 0: Abstract Let G be a group with identity e. A similar argument yields a similar contradiction and thus completes comltiplication proof. Let J be a proper graded ideal of R.

As a dual concept of gr -multiplication modules, graded comultiplication modules gr -comultiplication modules were introduced and studied by Ansari-Toroghy and Farshadifar [8]. If M is a gr – comultiplication gr – prime R – comultipljcationthen M is a gr – simple module.

Hence I is a gr -small ideal of R. Conversely, let N be a graded submodule of M. Then M is gr – uniform if and only if R is gr – hollow. Let R be a G – graded ring and M a gr – comultiplication R – module. Volume 8 Issue 6 Decpp. Let R be a gr – comultiplication ring and M a graded R – module. Let G be a group with identity e. See all formats and pricing Online. A respectful treatment of one another is important to us.

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

If M is a gr – faithful R – module, then for each proper graded ideal J of R0: Volume 7 Issue 4 Decpp. In this paper we will obtain comultilpication results concerning the graded comultiplication modules over a commutative graded ring. Volume 9 Issue 6 Decpp. Volume 12 Issue 12 Decpp. Volume 3 Issue 4 Decpp. About the article Received: Volume 6 Issue 4 Decpp. De Gruyter Online Google Scholar. Let R be a G -graded ring and M an R -module. BoxIrbidJordan Email Other articles by this author: So I is a gr -small ideal of R.

Recall kodules a G -graded ring R is said to be a gr -comultiplication ring if it is a gr -comultiplication R -module see [8]. Mosules graded R -module M is said to be gr – simple if 0 and M are its only graded submodules. Volume 13 Issue 1 Jan Proof Let N be a gr -second submodule of M.

Volume 1 Issue modulse Decpp. Let R be a G -graded ring and M a graded R -module. Proof Suppose first that N is a gr -small submodule of M. Therefore M is a gr -simple module.

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Therefore M is a gr -comultiplication module. Some properties of graded comultiplication modules. Volume 15 Issue 1 Janpp.

modkles Let K be a non-zero graded submodule of M. A graded R -module M is said to be gr – Artinian if satisfies the descending chain condition for graded submodules.

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A graded submodule N of a graded R -module M is said to be graded minimal gr – minimal comultip,ication it is minimal in the lattice of graded submodules of M. Since N is a gr -second submodule of Mby [ 8Proposition 3. Volume 5 Issue 4 Dec comultipliction, pp. By [ 8Theorem 3. In this case, N g is called the g – component of N.

Here we will study the class of graded comultiplication modules and obtain some further results which are dual to classical results on graded multiplication modules see Section 2.

Proof Suppose first that N is a gr -large submodule of M. Since M is a gr -comultiplication module, 0: By[ 8Lemma 3. Let R be comultipliation G – graded ring and M a graded R – module.