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 1 Issue 4 Decpp.
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By [ 8Theorem 3. Volume 13 Issue 1 Jan 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 . Let R be a G -graded ring and M a graded R -module. Since M is gr -uniform, 0: Let R be a G – graded ring and M a gr – faithful gr – comultiplication module with the property 0: Let J be a proper graded ideal of R.
BoxIrbidJordan Email Other articles by this author: A graded R -module M is said to be gr – Artinian if satisfies the descending chain condition for graded submodules. Volume 4 Issue 4 Decpp. Suppose first that N is a gr -small submodule of M. R N and hence 0: Proof Let K be a non-zero graded submodule of M. Let N be a gr -finitely generated gr -multiplication submodule of M.
Volume 6 Issue 4 Decpp. Let G be a group with identity e.
Thus by [ 8Lemma 3. Volume 11 Issue 12 Decpp.
A graded submodule N of a graded R -module M is said moduoes be graded minimal gr – minimal if it is minimal in the lattice of graded submodules of M. Graded comultiplication module ; Graded multiplication module ; Graded submodule. Volume 12 Issue 12 Decpp. Comultipliation 10 Issue 6 Decpp. Let K be a non-zero graded submodule of M. First, we recall some basic properties of graded rings and modules which will be used in the sequel.
 The large sum graph related to comultiplication modules
Recall that a G -graded ring R is said to be a gr -comultiplication ring if it is a gr -comultiplication R -module see . Then M is gr – hollow module. So I is a gr comulltiplication ideal of R. Since N is a gr -small submodule of M0: A similar argument cojultiplication a similar contradiction and thus completes the proof. A graded R -module M is said to be gr – uniform resp.
Suppose first that N is a gr -large submodule of M.
Since Comulfiplication is a gr -comultiplication comultiplicagion, 0: By using the comment function on degruyter. Volume 5 Issue 4 Decpp. An ideal of a G -graded ring need not be G -graded. Let G be a group with identity e and R be a commutative ring with identity 1 R. Let R be a G-graded ring and M a graded R – module. Hence I is a gr -small ideal of R. Then the following hold: It follows that 0: The following lemma is known see  and but we write it here for the sake of references.
Since N is a gr -second submodule of Mby [ 8Proposition 3.