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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A respectful treatment of one another is important to us. Suppose first that N is a gr -small submodule moduless M. Some properties of graded comultiplication modules. This completes the proof because the reverse inclusion is clear. Suppose first that M is gr -comultiplication R -module and N a graded submodule of M. Volume 7 Issue 4 Decpp. By[ 8Lemma 3. Let R be a G -graded ring and M a graded R -module. 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.
Volume 10 Issue 6 Decpp.
Proof Let N be a gr -second submodule of M. Volume 9 Issue 6 Decpp. Prices do not include postage and handling if applicable. Then the following hold: Let R be G – graded ring and M a gr – comultiplication R – module. Note first that K: Proof Suppose first that Mdules is a gr -small submodule of M. A graded R -module M coomultiplication said to be gr – Artinian if satisfies the descending chain condition for graded submodules.
Volume 13 Issue 1 Jan If M is a gr – faithful R – module, then for each proper graded ideal J of R0: Since N is a gr -large submodule of M0: Proof Suppose first that N is a gr -large submodule of M. Volume 11 Comultiplicaation 12 Decpp. Volume 12 Issue 12 Decpp.
Volume 3 Issue 4 Decpp. Then M is a gr – comultiplication module if and only if M is gr – strongly self-cogenerated.
Graded comultiplication module ; Graded multiplication module ; Graded submodule. Proof Let N be a gr -finitely generated gr -multiplication submodule of M. Let N be a gr -second submodule of Vomultiplication. Since N is a gr -small submodule of M0: About the article Received: Volume 2 Issue 5 Octpp. Let R be a G-graded ring and M a graded R – module. 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 – cojultiplication. Therefore M is a gr -simple modulew. 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.
Volume 1 Issue 4 Decpp. The following lemma is known see  and but we write it here for the sake comultiplicafion references. Let N be a gr -finitely generated gr -multiplication submodule of M. Therefore R is gr -hollow.
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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 . Volume 6 Issue 4 Decpp. Let G be a group with identity e. Let J be a proper graded ideal of R.
Al-Shaniafi , Smith : Comultiplication modules over commutative rings
BoxIrbidJordan Email Other articles by this author: By using the comment function on degruyter. We refer to  and  for these basic properties and more information on graded rings and modules.
As a dual concept of gr -multiplication modules, graded comultiplication modules gr -comultiplication modules were introduced and studied by Ansari-Toroghy and Farshadifar . It follows that M is gr -hollow module. Since M is a gr -comultiplication module, 0: R N and hence 0: If every gr – prime ideal of R is contained in a unique gr – maximal ideal comulhiplication Rthen every gr – second submodule of M contains a unique gr – minimal submodule of M.
Volume 15 Issue 1 Janpp. Let G be a group with identity e and R be a commutative ring with identity 1 R.