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 7 Issue 4 Decpp. Volume 8 Issue 6 Decpp. User Account Log in Register Help. By using the comment function on degruyter. An ideal of a G -graded ring need not be G -graded. Let R be a G -graded ring and M an R -module. Let J be a proper graded ideal of R. A similar argument yields a similar contradiction and thus completes the proof.

Proof Suppose first that N is a gr -large submodule of M. Therefore we would like to draw your attention to our House Rules.

Therefore M is a gr -simple module. Volume 6 Issue 4 Decpp.

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Volume 2 Issue 5 Octpp. It follows that 0: Volume 15 Issue 1 Janpp.

Since N is a gr -second submodule of Mby [ 8Proposition 3. A graded R -module M is said to be gr – Artinian if satisfies the descending chain condition for graded submodules.

Then the following hold: Thus by [ 8Lemma 3. Volume 4 Issue comulhiplication Decpp. Let G be a group with identity e.

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

Suppose first that N is a gr -small submodule of M. First, we recall some basic properties of graded rings and modules which will be used in the sequel. Conversely, let N be a graded submodule of M. If M is a gr – comultiplication gr – prime R – modulethen M is a gr – simple module. 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.

Graded multiplication modules gr -multiplication modules over commutative graded ring have been studied by many authors extensively see [ 1 — 7 ].

About the article Received: Since M is gr -uniform, 0: By[ 8Lemma 3. Therefore M is a gr -comultiplication module. In this case, N g is called the g – component of N.

Hence I is a gr -small ideal of R. Let R be G – graded ring and M a gr – comultiplication R – module. Since M is a gr -comultiplication module, 0: Proof Suppose first that N is a gr -small submodule of M. Let I be an ideal of R.

The following lemma is known see [12] and [6]but we write it here for the sake of references. Volume 3 Issue 4 Decpp. BoxIrbidJordan Email Other articles by this author: Volume 12 Issue 12 Decpp. Let R be a G – graded ring and M a gr – comultiplication R – module. De Gruyter Online Google Scholar. Volume 13 Issue 1 Jan This completes the proof because the reverse inclusion is clear. Let R be a G-graded ring and M a graded R – module. Since N is a gr -large submodule of M0: By [ 1Theorem 3.

Suppose first that N is a gr -large submodule of M. Since N is a gr -small submodule of M0: Proof Note first that K: Let R be a G – graded ring and M a graded R – module.