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 12 Issue 12 Decpp. R N and hence 0: Volume 6 Issue 4 Decpp.

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

Then M is gr – uniform if and only if R is gr – hollow. Since N is a gr -small submodule of M0: Since M is gr -uniform, 0: Then M is gr – hollow module. Let R be a G-graded ring and M a graded R – module. Let R be a G – graded ring and M a gr – comultiplication R – module. Then the following hold: First, we recall some basic properties of graded rings and modules which will be used in the sequel. Proof Suppose first that N is a gr -large submodule of M.

Volume 7 Issue 4 Decpp. It follows that M is gr -hollow module. Volume 8 Issue 6 Decpp. Proof Let N be a gr -second submodule of M.

Then M is a gr – comultiplication module if and only if M is gr – strongly self-cogenerated. Proof Let J be a proper graded ideal of R. Volume 5 Issue 4 Decpp. By [ 8Theorem 3. So I is a gr -small ideal of R. The following lemma is known see [12] and [6]but we write it here for the sake of references.

A similar argument yields a similar contradiction and thus completes the proof. Thus by [ 8Lemma 3. In this paper we will obtain some results concerning the graded comultiplication modules over a commutative graded ring. Since M is a gr -comultiplication module, 0: Conversely, let N be a graded submodule of M.

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. My Content 1 Recently viewed 1 Some properties of gra By using the comment function on degruyter. Let G be a group with identity e. Let R be a G – graded ring and M a graded R – module.

## Open Mathematics

Prices do not include postage and handling if applicable. Let R be a G -graded ring and M an R -module.

Volume 9 Issue 6 Decpp.