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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If M is a gr – comultiplication gr – prime R – modulethen M is a gr – simple module. Let N be a gr -finitely generated gr -multiplication submodule of M. By [ 8Theorem 3. About the article Received: Volume 11 Issue 12 Decpp.
Proof Suppose mdoules that N is a gr -small mmodules of M. 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: Volume 2 Issue 5 Octpp. Since N is a gr -second submodule of Mby [ 8Proposition 3.
Let R be a G -graded commutative ring and M a graded R -module. The following lemma is known see  and but we write it here for the sake of references. Let R be a G-graded ring and M a graded R – module. Proof Let J be a proper graded ideal of R.
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Volume 13 Issue 1 Jan By [ 1Theorem moddules. Proof Suppose first that N is a gr -large submodule of M.
comultiplidation Since N is a gr -small submodule of M0: As a dual concept of gr -multiplication modules, graded comultiplication modules gr -comultiplication modules were introduced and studied by Ansari-Toroghy and Farshadifar . Then M is a gr – comultiplication module if and only if M is gr – strongly self-cogenerated.
Let G be a group with identity e. Volume 12 Issue modulles Decpp. Prices do not include postage and handling if applicable. Abstract Let G be a group with identity e. Prices are subject to change without notice. Volume 5 Issue 4 Decpp. Let N be a gr -second submodule of M.
This completes the proof because the reverse inclusion modiles clear. Note first that K: Let G be a group with identity e and R be a commutative ring with identity 1 R. Volume 6 Issue 4 Decpp. Suppose first that N is a gr -small submodule of M. Let I be an ideal of R.
Some properties of graded comultiplication modules : Open Mathematics
Therefore R is gr -hollow. Proof Let N be a gr -second submodule of M. A respectful treatment of one another is important to us. Therefore M is gr -uniform. A similar argument yields a similar contradiction and thus completes the proof. Suppose first that M is gr -comultiplication R -module and N a graded submodule of M.
By[ 8Lemma 3. Then M is gr – hollow module. Then M is gr – uniform if and only if R is gr – hollow. A graded R -module M is said to be gr – simple if 0 and M are its only graded submodules. Some properties of graded comultiplication modules.
First, we recall some basic properties of graded rings and modules which will be used in the sequel. By using the comment function on degruyter. Let K be a non-zero graded submodule of M. Since M is gr -uniform, 0: Hence I is a gr -small ideal of R. Let R be a G -graded ring and M a graded R -module. It follows that 0: Let R be a G – graded ring and M a graded R – module. Graded comultiplication module ; Graded multiplication module ; Graded submodule.
Volume 3 Issue 4 Decpp. Proof Note first that K: