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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Let G be a group with identity e. Let R be a G -graded commutative ring and M a graded R -module. In this paper we will obtain some results concerning the graded comultiplication modules over a commutative graded ring. Graded comultiplication module ; Graded multiplication module ; Graded submodule. Graded multiplication modules gr -multiplication modules over commutative graded ring have been studied by many authors extensively see [ 1 — 7 ].

As a dual concept of gr -multiplication modules, graded comultiplication modules comultiplicatikn -comultiplication modules were introduced and studied by Ansari-Toroghy and Farshadifar [8]. Here we will study the class of graded comultiplication modules and obtain some further results which are dual to modyles results on graded multiplication modules see Section 2. First, we recall some basic properties of graded rings and modules which will be used in the sequel.

We refer to [9] and [10] for these basic properties and more information on graded rings and modules. Let G be a group with identity e and R be a commutative ring with identity 1 R. Let I be an ideal of R. An ideal mosules a G -graded ring need not be Comultiplicxtion -graded. Let R be a G -graded ring and M an R comultiplicationn.

In this case, N g is called cmultiplication g – component of N. Let R be a G -graded ring and M a graded R -module. A graded submodule N of a graded R -module M conultiplication said to be graded minimal gr – minimal if it is minimal in the lattice of graded submodules of M. A graded R -module M is said to be gr – uniform resp. A graded R -module M is said comultiplicaion be gr – simple if 0 and M are its only graded submodules.

A graded R -module M is said to be gr – Artinian if satisfies the descending chain condition for graded submodules. A non-zero graded submodule N of a graded R -module M is said to be a graded second gr – second if for each homogeneous element a of Rthe endomorphism of M given by multiplication by a is either surjective or zero see [8]. The following lemma is known see [12] and [6]but we write it here for the sake of references.

Let R be a G – graded ring and M a graded R – module. Then the following hold: If M comultpilication a gr – comultiplication gr – prime R – modulethen M is a gr – simple module. Let K be a non-zero graded submodule of M. By[ 8Lemma 3. Therefore M is a gr -simple module.

Let R be a G-graded ring and M a graded R – module. This completes the proof because the reverse inclusion is clear. 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]. Let R be a gr – comultiplication ring and M a graded R – module. If M is a gr – faithful R – module, then for each proper graded ideal J of R0: Let J be a proper graded ideal of R.

A similar argument yields a similar contradiction and thus completes the proof. By [ 8Theorem 3. Let R be G – graded ring and M a gr – comultiplication R – module.

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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 N be a gr -second submodule of M. Since N is a gr -second submodule of Mby [ 8Proposition 3. Thus by [ 8Lemma 3. Let R be a G – graded ringM a gr – comultiplication R – module and 0: Let N be a gr -finitely generated gr -multiplication submodule of M. Note first that K: Since M is a gr -comultiplication module, 0: R N and hence 0: Let R be a G – graded ring and M a gr – comultiplication R – module.

By [ 1Theorem 3. Then M is gr – hollow module.

## Mathematics > Commutative Algebra

It follows that M is gr -hollow module. Let R be a G – graded ring and M a gr – faithful gr – comultiplication module with the property 0: Suppose first that N is a gr -small submodule of M. Since N is a gr -small submodule of M0: Thus I is a gr -large ideal of R.

Suppose first that N is a gr -large submodule of M. Since N is a gr -large submodule mldules M0: So I is a gr -small ideal of R. It follows that 0: Then M is gr – uniform if and only if R is gr – hollow. Since M is gr -uniform, 0: Hence I is a gr -small ideal of R. Therefore R is gr -hollow.

Therefore M comultipliication gr -uniform. Then Modulew is a gr – comultiplication module if and only if M is gr – strongly self-cogenerated.

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

Suppose first that M is gr -comultiplication R -module and N a graded submodule of M. Conversely, let N be a graded submodule of M. Therefore M is a gr -comultiplication module. User Account Log in Register Help. My Content 1 Recently viewed 1 Some properties of gra See all formats and pricing Online. Prices are subject to change without notice. Prices do not include postage and handling if applicable. Volume 15 Issue 1 Janpp.

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### Al-Shaniafi , Smith : Comultiplication modules over commutative rings

Volume 1 Issue 4 Decpp. Some properties of graded comultiplication modules. BoxIrbidJordan Email Other articles by this author: De Gruyter Online Google Scholar. Abstract Let G be a group with identity e.

Proof Let K be a non-zero graded submodule of M. Proof Let J be a proper graded ideal of R.

Proof Let N be a gr -second submodule of M. Proof Let N be a gr -finitely generated gr -multiplication submodule of M.

Proof Note first that K: Proof Suppose first that N is a gr -small submodule of M. Proof Suppose first that N is a gr -large submodule of M. About the article Received: By using the comment function on degruyter.