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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An ideal of a G -graded ring need not be G -graded. Then the following hold: If M is a gr – faithful R – module, then for each proper graded ideal J of R0: See all formats and pricing Online. Volume 3 Issue 4 Decpp. Let R be a G – graded ring and M a graded R – module. Let N be a gr -finitely generated gr -multiplication submodule of M. Let G be a group with identity e and R be a commutative ring with identity 1 R.

De Gruyter Online Google Scholar. Volume 1 Issue 4 Decpp.

### On semiprime comultiplication modules over pullback ringsAll

Proof Suppose first that N is a gr -small submodule of M. Suppose first that N is a gr -small submodule of M. BoxIrbidJordan Email Other articles by this author: The following lemma is known see [12] and [6]but we write it here for the sake of references.

A graded R -module M is said to be gr – Artinian if satisfies the descending chain condition for graded submodules. Let R be a G -graded ring and M a graded R -module. My Content 1 Recently viewed 1 Some properties of gra So I is a gr -small ideal of R. Let K be a non-zero graded submodule of M. Therefore R is gr -hollow. A similar argument yields a similar contradiction and thus completes the proof.

Volume 5 Issue 4 Decpp. Here we will study the class of graded comultiplication modules and obtain some further results which are dual to classical results on graded multiplication modules see Section 2.

Volume 13 Issue 1 Jan Conversely, let N be a graded submodule of M. User Account Log in Register Help. A respectful treatment of one another is important to us.

By [ 8Theorem 3.

## Mathematics > Commutative Algebra

Then M is gr – hollow module. Let G be a group with identity e. Volume 10 Issue 6 Decpp. Since N is a gr -small submodule of M0: If M is a gr – comultiplication gr – prime R – modulethen M is a gr – simple module. Note first that K: Proof Let K be a non-zero graded submodule of M.

Prices do not include postage and handling if applicable. Since M is a gr -comultiplication module, 0: Volume 4 Issue 4 Decpp. Let R be G – graded ring and M a gr – comultiplication R – module. Volume 14 Issue 1 Janpp. Therefore M is a gr -comultiplication module.

Suppose first that N is a gr -large submodule of M. Therefore we would like to draw your attention to our House Rules. A non-zero graded submodule Comultiplicatiob of a graded R -module M is said to be a graded second gr – second if for each homogeneous element a of Rthe mmodules of M given by multiplication by a is either surjective or zero see [8].

Let J be a proper graded ideal of R. Since N is a gr -large comultiplicationn of M0: Let N be a gr -second submodule of M. By [ 1Theorem 3. Since M is gr -uniform, 0: This completes the proof because the reverse inclusion is clear. By[ 8Lemma 3. Proof Let N be a gr -finitely generated gr -multiplication submodule of M.

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

Volume 12 Issue 12 Decpp. As a dual concept of gr -multiplication modules, graded comultiplication modules gr -comultiplication modules were introduced and studied by Ansari-Toroghy and Farshadifar [8]. Proof Let N be a xomultiplication -second submodule of M. Volume 2 Issue 5 Octpp.