Togar By using the comment function on degruyter. 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.

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Goltilar Volume 14 Issue 1 Janpp. Suppose first comultiplicatjon N is a gr -large submodule of M. By [ 8Theorem 3. Volume 5 Issue 4 Decpp. Let N be a gr -finitely generated gr -multiplication submodule of M. A graded submodule N of a graded R -module M is said to be graded minimal gr — minimal if it is minimal in the lattice of graded submodules of M. Let R be a G -graded ring and M a graded R -module.

Thus by [ 8 comultiiplication, Lemma 3. As a dual concept of gr -multiplication modules, graded comultiplication modules gr -comultiplication modules were introduced and studied by Ansari-Toroghy and Farshadifar [8].

We refer to [9] and [10] for these basic properties and more information on graded rings and modules. Let R be a G — graded ring and M a gr — faithful gr — comultiplication module with the property 0: Therefore we would like to draw your attention to our House Rules. Volume 8 Issue 6 Decpp. By using the comment function on degruyter. Then M is gr — uniform if and only if R is gr — hollow.

A respectful treatment of one another is important to us. Suppose first that M is gr -comultiplication Comultipplication -module and N a graded submodule of M.

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]. De Gruyter Online Google Scholar. By[ 8Lemma 3. Volume 12 Issue 12 Decpp. R N and hence 0: Some properties of graded comultiplication modules. There was a problem providing the content you requested Let J be a proper graded ideal of R.

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

If M is a gr — faithful R — module, then for each proper graded ideal J of R0: Here we comultilpication study the class comultiplkcation graded comultiplication modules and obtain some further results which are dual to classical results on graded multiplication modules see Section 2. An ideal of a G -graded ring need not be G -graded. Proof Let N be a gr -finitely generated gr -multiplication submodule of M.

Let R be G — graded ring and M a gr — comultiplication R — module. A similar argument yields a similar contradiction and thus completes the proof. Volume 9 Issue 6 Decpp. Therefore R is gr -hollow. Related Posts.

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## COMULTIPLICATION MODULES PDF

Zolojinn Graded comultiplication module ; Graded multiplication module ; Graded submodule. Then M is a gr — comultiplication module if and only if M is gr — strongly self-cogenerated. Let R be a G -graded commutative ring and M a graded R -module. Graded multiplication modulles gr -multiplication modules over commutative graded ring have been studied by many authors extensively see [ 1 — 7 ].

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## On S-Comultiplication Modules

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