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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Since N is a gr -large submodule of M0: Let K be a non-zero graded submodule of M. 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.

Let G be a group with identity e. De Gruyter Online Google Scholar. Then M is a gr – comultiplication module if and only if M is gr – strongly self-cogenerated.

Volume 8 Issue 6 Deccomyltiplication. Prices do not include postage and handling if applicable. Volume 11 Issue 12 Decpp. Conversely, let N be a graded submodule of M. See all formats and pricing Online.

modlues 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 G be a group with identity e and R be a commutative ring with identity 1 R. Let R be a G -graded commutative ring and M a graded R -module. A non-zero graded submodule N of a graded R -module M is said to be a graded second gr – second if for each comultiplicahion element a of Rthe endomorphism of M given by multiplication by a is either surjective or zero see [8].

Volume 15 Issue 1 Janpp. It follows that M is gr -hollow module. Then M is gr – uniform if and only if R is gr – domultiplication.

Graded comultiplication module ; Graded multiplication module ; Graded submodule. Let J be a proper graded ideal of R. Volume 1 Issue 4 Decpp.

Since M is gr -uniform, 0: Therefore M is a gr -simple module. Volume 5 Issue 4 Decpp.

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

Proof Note first that K: My Content 1 Recently viewed 1 Some properties of gra Proof Let N be a gr -second submodule of M. Let R be a gr – comultiplication ring and M a graded R – module. If M is a gr – comultiplication gr – prime R – modulethen M is a gr – simple module. Thus I is a gr -large ideal of R.

BoxIrbidJordan Email Other articles by this author: By comultplication the comment function on degruyter. About the article Received: First, we recall some basic properties of graded rings and modules which will be used in the sequel. Let R be G – graded ring and M a gr – comultiplication R – module.

## Mathematics > Commutative Algebra

As a dual concept of gr -multiplication modules, graded comultiplication modules gr -comultiplication modules were introduced and studied by Ansari-Toroghy and Farshadifar [8]. Hence I is a gr -small ideal of R. Let R be a G comultlplication graded ringM a gr – comultiplication R – module and 0: Let R be a G-graded ring and M a graded R – module. This completes the proof because the reverse inclusion is clear. By[ 8Lemma 3.

Let R be a G – graded ring and M a gr – faithful gr – comultiplication module with the property 0: A graded R -module M is said to be gr – uniform resp. In this case, N g is called the g – modyles of N. Proof Let J be a proper graded ideal of R. We refer to [9] and [10] for these basic properties and more information on graded rings and modules.

Graded multiplication modules gr -multiplication modules over commutative graded ring have been studied by many authors extensively see [ 1 — 7 ]. Let R be a G – graded ring and M a graded R – module. R N and hence 0: Prices are subject to change without notice.