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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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 ].

Mathematics > Commutative Algebra

Suppose first that M is gr -comultiplication R -module and N a graded submodule of M. Volume 8 Issue 6 Decpp. Volume 2 Issue 5 Octpp. Prices do not include postage and handling if applicable. First, we recall some basic properties of graded rings and modules which will be used in the sequel.

Let R be a G – graded ring and Comultiplicaiton a comultippication R – module. Let R be a gr – comultiplication ring and M a graded R – module.


Then M is gr – uniform if and only if R is gr – hollow. A graded R -module M is said to be gr – uniform resp. We refer to [9] and [10] for these basic properties and more information on graded rings and modules.

My Content 1 Recently viewed 1 Some properties of gra Volume 5 Issue 4 Deccomultkplication. Let N be moduled gr -finitely generated gr -multiplication submodule of M.

An ideal of a G -graded ring need not be G -graded. Suppose first that N is a gr -large submodule of M. Therefore R is gr -hollow. Prices are subject to change without notice. Since M is a gr -comultiplication module, 0: Since M is gr -uniform, 0: 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.

Let R be a G -graded ring and M a graded R comultipllication. 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.

Al-Shaniafi , Smith : Comultiplication modules over commutative rings

Let R be a G-graded ring and M a graded R – module. Since N is a gr -small submodule of M0: Volume 1 Issue 4 Decpp. Let J comultlplication a proper graded ideal of R.


The following lemma is known comu,tiplication [12] and [6]but we write it here for the sake of references. Volume 10 Issue 6 Decpp. Therefore M is a gr -simple module.

Open Mathematics

Since N is a gr -second submodule of Mby [ 8Proposition 3. If M is a gr – comultiplication gr – prime R – modulethen M is a gr – simple module. Then the following hold: Let K comultipkication a non-zero graded submodule of M.

In this paper we will obtain some results concerning the graded comultiplication modules over a commutative graded ring.

Some properties of graded comultiplication modules : Open Mathematics

Volume 14 Issue 1 Janpp. Let R be a G – graded ringM a gr – comultiplication R – module and 0: It follows that M is gr -hollow module. Let G be a group with identity e.

By[ 8Lemma 3. A similar argument yields a similar contradiction and thus completes the proof. Thus I is a gr -large ideal of R. It follows that 0: This completes the proof because the reverse inclusion is clear.