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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Suppose first that N is a gr -small submodule of M.

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

User Account Log in Register Help. We refer to [9] and [10] for these basic properties and more information on graded rings and modules. Abstract Let G be a group with identity e.

Prices are subject to change without notice. By [ 8Theorem 3. Let R be a G -graded commutative ring and M a graded R -module. Since M is gr -uniform, 0: De Gruyter Online Google Scholar. Proof Suppose first that N is a gr -large submodule of M. A graded R -module M is said to be gr – uniform resp.

Proof Note first that K: Since M is a gr -comultiplication module, 0: Then M is gr – hollow module. Let R be G – graded ring and M a gr – comultiplication R – module. 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]. Volume 1 Issue 4 Decpp. Let R be a G – graded ring and M a graded R – module. Therefore we would like to draw your attention to our House Rules.

Then M is a gr – comultiplication module if and only if M is gr – strongly self-cogenerated. Let I be an ideal of R. Volume 4 Issue 4 Decpp. Therefore M is a gr -comultiplication module.

Volume 11 Issue 12 Decpp. First, we recall some basic properties of graded rings and modules which will be used in the sequel. Proof Let J be a proper graded ideal of R. Proof Let N be a gr -second submodule of M. Hence I is a gr -small ideal of R.

By[ 8Lemma 3. By using the comment function on degruyter. The following lemma is known see [12] and [6]but we write it here for the sake of references.

Suppose first that N is a gr -large submodule of M. Thus I is a gr -large ideal of R. 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.

Proof Let K be a non-zero graded submodule of M. Let G be a group with identity e and R be a commutative ring with identity 1 R. Prices do not include postage and handling if applicable.

### On semiprime comultiplication modules over pullback ringsAll

Therefore M is a gr -simple module. Thus by [ 8Lemma 3. Let N be a gr -finitely generated gr -multiplication submodule of M. Let N be a gr -second submodule of M.

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Volume 6 Issue 4 Decpp. As a dual concept of gr -multiplication modules, xomultiplication comultiplication modules gr -comultiplication modules were introduced and studied by Ansari-Toroghy and Farshadifar [8].

Volume 5 Issue 4 Decpp. Then the following hold: Since N comultiplicatiob a gr -large submodule of M0: 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].

A graded R -module M is said to be gr – Artinian if satisfies the descending chain condition for graded submodules. Proof Let N be a gr -finitely generated gr -multiplication submodule of M. Volume 2 Issue 5 Octpp. BoxIrbidJordan Email Other articles by this author: Therefore R is gr -hollow. An ideal of a G -graded ring need not be G -graded. About the article Received: Graded comultiplication module ; Graded multiplication module ; Graded submodule. Since N is a gr -second submodule of Mby [ 8Proposition 3.

Volume 12 Issue 12 Decpp.