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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Suppose first that N is a gr -small submodule of M.
 The large sum graph related to comultiplication modules
User Account Log in Register Help. We refer to  and  for these basic properties and more information on graded rings and modules. Abstract Let G be a group with identity e.
Prices are subject to change without notice. By [ 8Theorem 3. Let R be a G -graded commutative ring and M a graded R -module. Since M is gr -uniform, 0: De Gruyter Online Google Scholar. Proof Suppose first that N is a gr -large submodule of M. A graded R -module M is said to be gr – uniform resp.
Proof Note first that K: Since M is a gr -comultiplication module, 0: Then M is gr – hollow module. Let R be G – graded ring and M a gr – comultiplication R – module. Recall that a G -graded ring R is said to be a gr -comultiplication ring if it is a gr -comultiplication R -module see . Volume 1 Issue 4 Decpp. Let R be a G – graded ring and M a graded R – module. Therefore we would like to draw your attention to our House Rules.
Then M is a gr – comultiplication module if and only if M is gr – strongly self-cogenerated. Let I be an ideal of R. Volume 4 Issue 4 Decpp. Therefore M is a gr -comultiplication module.
Volume 11 Issue 12 Decpp. First, we recall some basic properties of graded rings and modules which will be used in the sequel. Proof Let J be a proper graded ideal of R. Proof Let N be a gr -second submodule of M. Hence I is a gr -small ideal of R.
By[ 8Lemma 3. By using the comment function on degruyter. The following lemma is known see  and but we write it here for the sake of references.
Suppose first that N is a gr -large submodule of M. Thus I is a gr -large ideal of R. 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.
Proof Let K be a non-zero graded submodule of M. Let G be a group with identity e and R be a commutative ring with identity 1 R. Prices do not include postage and handling if applicable.
On semiprime comultiplication modules over pullback ringsAll
Therefore M is a gr -simple module. Thus by [ 8Lemma 3. Let N be a gr -finitely generated gr -multiplication submodule of M. Let N be a gr -second submodule of M.
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Volume 6 Issue 4 Decpp. As a dual concept of gr -multiplication modules, xomultiplication comultiplication modules gr -comultiplication modules were introduced and studied by Ansari-Toroghy and Farshadifar .
Volume 5 Issue 4 Decpp. Then the following hold: Since N comultiplicatiob a gr -large submodule of M0: 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 .
A graded R -module M is said to be gr – Artinian if satisfies the descending chain condition for graded submodules. Proof Let N be a gr -finitely generated gr -multiplication submodule of M. Volume 2 Issue 5 Octpp. BoxIrbidJordan Email Other articles by this author: Therefore R is gr -hollow. An ideal of a G -graded ring need not be G -graded. About the article Received: Graded comultiplication module ; Graded multiplication module ; Graded submodule. Since N is a gr -second submodule of Mby [ 8Proposition 3.
Volume 12 Issue 12 Decpp.