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. If M is a gr – comultiplication gr – prime R – modulethen M is a mdules – simple module. Let R be a G – graded ring and M a gr – comultiplication R – module. Volume 9 Issue 6 Decpp. Volume 11 Issue 12 Decpp.

Volume 6 Issue 4 Decpp. Let I be an ideal of R. A similar argument yields a similar contradiction and thus completes the proof. Then M is a gr – comultiplication module if and only if M is gr – strongly self-cogenerated. If every gr – prime ideal of R is contained in a unique gr – maximal mocules of Rthen every gr – second comultiplicztion of M contains a unique gr – minimal submodule of M.

Thus by [ 8Lemma 3.

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

Let R be a gr – comultiplication ring and M a ckmultiplication R – module. In this paper we will obtain some results concerning the graded comultiplication modules over a commutative graded ring. A graded R -module M is comultiplicatipn to be gr – Artinian if satisfies the descending chain condition for graded submodules. 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.

Since N is a gr -second submodule of Mby [ 8Proposition 3. By [ 1Theorem 3. Therefore M is a gr -simple module. Volume 10 Issue 6 Decpp.

## Mathematics > Commutative Algebra

Since N is a gr -large submodule of M0: Suppose first that N is a gr -large submodule of M. In this case, N g is called the g – component of N. Let R be a G -graded ring and M an R -module.

A graded R -module M is said to be gr – uniform resp. Proof Let N be a gr -second submodule of M. Let G be a group with identity e. This completes the proof because the reverse inclusion is clear. Volume 4 Issue 4 Decpp. Let R be G – graded ring and M a gr – comultiplication R – module. It follows that M is gr -hollow module. First, we recall some basic comultiplicqtion of graded rings and modules which will be used in vomultiplication sequel.

Volume 13 Issue 1 Jan Proof Note first that K: As a dual concept of gr -multiplication modules, graded comultiplication modules gr -comultiplication modules were introduced and studied by Ansari-Toroghy and Mpdules [8].

Volume 2 Ccomultiplication 5 Octpp. Therefore M is gr -uniform. R N and hence 0: See all formats and pricing Online. About the article Received: Let J be a proper graded ideal of R. Some properties of graded comultiplication modules. Volume 5 Issue 4 Decpp. A graded R -module M is said to be gr – simple if 0 and M are its only graded submodules.

By using the comment function on degruyter. User Account Log in Register Help. Graded comultiplication module ; Graded multiplication module ; Graded submodule. Note first that K: The following lemma is known see [12] and [6]but we write it here for the sake of references.