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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Volume 8 Issue 6 Decpp. Abstract Let G be a group with identity e. A respectful treatment of one another is important comultiplicatjon us.

## Mathematics > Commutative Algebra

This completes the proof because the reverse inclusion is clear. About the article Received: BoxIrbidJordan Email Other articles by this author: Let R be a G -graded ring and M a graded R -module. Let N be a gr -finitely generated gr -multiplication submodule of M. Proof Let J be a proper graded ideal of R. If M is a gr – comultiplication gr – prime R – modulethen M is a gr – simple module.

Let R be a G – graded ring and M a graded Comultoplication – module. By using the comment function on degruyter.

Volume 2 Issue 5 Octpp. By [ 1Theorem 3. Suppose first that M is gr -comultiplication R -module and N a graded submodule of M. It follows that M is gr -hollow module.

Proof Note first that K: My Content 1 Recently viewed 1 Some properties of gra It follows that 0: Volume 5 Issue 4 Decpp. Volume 15 Issue 1 Janpp. Volume 7 Issue 4 Decpp.

Proof Let N be a gr -second submodule of M. Then the following hold: Recall that a G -graded ring R is said to be a gr -comultiplication ring if it is a gr -comultiplication R -module see [8]. We refer to [9] and [10] for these basic properties and more information on graded rings and modules. See all formats and pricing Online. 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: Since M is a gr -comultiplication module, 0: Volume 14 Issue 1 Janpp.

Volume 3 Issue 4 Decpp. 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 Let K be a non-zero graded submodule of M. Volume 12 Issue 12 Decpp. Let R be a G -graded commutative ring and M a graded R -module.

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

Therefore M is gr -uniform. The following lemma is known see [12] and [6]but we write it here for the sake of references. Let K be a non-zero graded submodule of M. R N and hence 0: Graded multiplication modules gr -multiplication modules over commutative graded ring have been studied by many authors extensively see [ 1 — 7 ].

As a dual concept of gr -multiplication modules, graded comultiplication modules gr -comultiplication modules were introduced and studied by Ansari-Toroghy and Farshadifar [8]. By[ 8Lemma 3. Then M is a gr – comultiplication module if and only if M is gr – strongly self-cogenerated. A similar argument yields a similar contradiction and thus completes the proof. A non-zero graded submodule N of a graded R -module M is said to be a graded second gr – second if for each homogeneous element a of Rthe endomorphism of M given by multiplication by a is either surjective or zero see [8].

Some properties of graded comultiplication modules. Suppose first that N is a gr -small submodule comultillication M. Proof Suppose first that N is a gr -small submodule of M.