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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Let N be a gr -finitely generated gr -multiplication submodule of M.
Volume 6 Issue 4 Decpp. Let R be a G -graded ring and M a graded R -module. Since M is gr -uniform, 0: Let G be a group with identity movules and R be a commutative ring with identity 1 R. User Account Log in Register Help. Volume 1 Issue 4 Decpp. The following lemma is known see  and but we write it here for the sake of references.
Therefore M is gr -uniform.
On semiprime comultiplication modules over pullback ringsAll
Proof Let J be a proper graded ideal of R. Let G be a group with identity e. Volume 8 Issue 6 Decpp. BoxIrbidJordan Email Other articles by this author: An ideal of a G -graded ring need not be G -graded. A graded Comultiplixation -module M is said to be gr – uniform resp.
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 . Proof Let N be a gr -second submodule of M.
Let R be a G -graded commutative ring and M a graded R -module. Volume 11 Issue 12 Decpp.
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Proof Let N be a gr -finitely generated gr -multiplication submodule of M. Conversely, let N be a graded submodule of M. Thus by [ 8Lemma 3. If M is a gr – faithful R – module, then for each proper graded ideal J of R0: Let R be a G – graded ringM a gr – comultiplication R – module and 0: Therefore M is a gr -simple module.
Let R be a gr – comultiplication ring and M a graded R – module. Volume 14 Issue 1 Janpp. So I is a gr -small ideal of R.
By using the comment function on degruyter. Proof Let K be a non-zero graded submodule of M. Comultiplicatio 2 Issue 5 Octpp. Volume 13 Issue 1 Jan Let I be an ideal of R. Volume 3 Issue 4 Decpp. Volume 15 Issue 1 Janpp. In this case, N g is called the g – component of N. Volume 5 Issue 4 Decpp. 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 mdules by a is either surjective or zero see .
Then M is gr – hollow module. Let K be a non-zero graded submodule of M. Some properties of graded comultiplication modules. It follows that M is gr -hollow module. Let R be a G – graded ring and M a gr – faithful gr – comultiplication module with the property 0: See all formats and pricing Online. De Gruyter Online Google Scholar. Hence I is a gr -small ideal of R. Let N be a gr -second submodule of M.
Let R be a G-graded ring and M a graded R – module. 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.
Al-Shaniafi , Smith : Comultiplication modules over commutative rings
Volume 7 Issue 4 Decpp. If M is a gr – comultiplication gr – prime R – modulethen M is a gr – simple module.
R N and hence 0: Since N is a gr -large submodule of M0: Volume 12 Issue 12 Decpp. Volume 10 Issue 6 Decpp. Graded comultiplication module ; Graded multiplication module ; Graded submodule. Thus I is a gr -large ideal of R.