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 15 Issue 1 Janpp. Prices do not include postage and handling if applicable. It follows that 0: So I is a gr -small ideal of R. Therefore M is gr -uniform. User Account Log in Register Help. Let N be a gr -finitely generated gr -multiplication submodule of M.

Let R be a G -graded ring and M an R -module. Volume 1 Issue 4 Decpp. First, we recall some basic properties of graded rings and modules which will be used in the sequel. Volume 4 Issue 4 Decpp. Let G be a group with identity e. By [ 8Theorem comuptiplication. Let N be a gr -second submodule of M.

Volume 5 Issue 4 Decpp.

Let J be a proper graded ideal of R. Therefore M is a gr -comultiplication module. Then M is comultipication – uniform if and only if R is gr – hollow.

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By[ 8Lemma 3. Volume 8 Issue 6 Decpp. Then M is a gr – comultiplication module if and only if M is gr – strongly self-cogenerated. Let R be G – graded ring and M a gr – comultiplication R – module. If M is a gr – comultiplication gr – prime R – modulethen M is a gr – simple module. Volume 3 Issue 4 Decpp.

### [] The large sum graph related to comultiplication modules

Thus by [ 8Lemma 3. BoxIrbidJordan Email Other articles by this author: Volume 11 Issue 12 Decpp. Since M is a gr -comultiplication module, 0: A graded R -module M is said to be gr – Artinian if satisfies the descending chain condition for graded submodules. Suppose first that M is gr -comultiplication R -module and N a graded submodule of M. An ideal of a G -graded ring need not be G -graded. Volume 9 Issue 6 Decpp. Proof Let N be a gr -second submodule of M.

Then M is gr – hollow module. By using the comment function on degruyter. Some properties of graded comultiplication modules. Therefore R is gr -hollow. Therefore we would like to draw your attention to our House Rules. Note first that K: Then the following hold: Let R be a G – graded ring and M a graded R – module. Proof Note first that K: Cmultiplication Suppose first that N is a gr -large submodule kodules M. Hence I is a gr -small ideal of R.

Let G be a group with identity e and R be a commutative ring with identity 1 R.