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

### Some properties of graded comultiplication modules : Open Mathematics

Thus I is a gr -large ideal of R. Let N be a gr -finitely generated gr -multiplication submodule of M. Then M is a gr – comultiplication module if and only if M is gr – strongly self-cogenerated. Volume 13 Issue 1 Jan Since M is a gr -comultiplication module, 0: A similar argument yields a similar contradiction and thus completes the proof. 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.

Thus by [ 8Lemma 3.

Suppose first that M is gr -comultiplication R -module and N a graded submodule of M. Let R be a G -graded commutative ring and M a graded R -module.

User Account Log in Comultiplciation Help. By[ 8Lemma 3.

By [ 8Theorem 3. A respectful treatment of one another is important to us.

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Then M is gr – hollow module. Proof Let K be a non-zero graded submodule of M. Volume 3 Issue 4 Decpp. BoxIrbidJordan Email Other articles by this author: Let R be G – graded ring and M a gr – comultiplication R – module. Let G be a group with identity e. 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]. Therefore M is a gr -comultiplication module. A graded R -module M is said to be gr – simple if 0 and M are its only graded submodules.

Proof Suppose first modulss N is a gr -small submodule of M. Hence I is a gr -small ideal of R. So I is a gr -small ideal of R. First, we recall some basic properties of graded rings and modules which will be used in the sequel. Proof Note first that K: Volume 4 Issue 4 Decpp. Prices do not include postage and handling if applicable. By modulrs 1Theorem 3.

## Mathematics > Commutative Algebra

Prices are subject to change without notice. Since M is gr -uniform, 0: Let R be a G – graded ring and M a gr – comultiplication R – module. Volume 8 Issue 6 Decpp. Mocules 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].

Therefore R is gr -hollow.

We refer to [9] and [10] for these basic properties and more information on graded rings and modules. Graded comultiplication module ; Graded multiplication module ; Graded submodule. Some properties of graded comultiplication modules. Volume 7 Issue 4 Decpp. Then M is gr – uniform if and only if R is gr – hollow. Since N is a gr -large submodule of M0: Let R be a G – graded ring and M a gr – faithful gr – comultiplication module with the property 0: Let R be a G -graded ring and M an R -module.

Volume 6 Issue 4 Decpp.