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Title

Annihilating submodule graph for modules

Year: 1397
COI: JR_COMB-7-1_001
Language: EnglishView: 39
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Authors

Saeed Safaeeyan - Department of mathematical Sciences, Yasouj university,Yasouj, ۷۵۹۱۸-۷۴۸۳۱, IRAN.

Abstract:

Let R be a commutative ring and M an‎ ‎R-module‎. ‎In this article‎, ‎we introduce a new generalization of‎ ‎the annihilating-ideal graph of commutative rings to modules‎. ‎The‎ ‎annihilating submodule graph of M‎, ‎denoted by \Bbb G(M)‎, ‎is an‎ ‎undirected graph with vertex set \Bbb A^*(M) and two distinct‎ ‎elements N and K of \Bbb A^*(M) are adjacent if N*K=۰‎. ‎In‎ ‎this paper we show that \Bbb G(M) is a connected graph‎, ‎{\rm‎ ‎diam}(\Bbb G(M))\leq ۳‎, ‎and {\rm gr}(\Bbb G(M))\leq ۴ if \Bbb‎ ‎G(M) contains a cycle‎. ‎Moreover‎, ‎\Bbb G(M) is an empty graph‎ ‎if and only if {\rm ann}(M) is a prime ideal of R and \Bbb‎ ‎A^*(M)\neq \Bbb S(M)\setminus \{۰\} if and only if M is a‎ ‎uniform R-module‎, ‎{\rm ann}(M) is a semi-prime ideal of R‎ ‎and \Bbb A^*(M)\neq \Bbb S(M)\setminus \{۰\}‎. ‎Furthermore‎, ‎R‎ ‎is a field if and only if \Bbb G(M) is a complete graph‎, ‎for‎ ‎every M\in R-{\rm Mod}‎. ‎If R is a domain‎, ‎for every divisible‎ ‎module M\in R-{\rm Mod}‎, ‎\Bbb G(M) is a complete graph with‎ ‎\Bbb A^*(M)=\Bbb S(M)\setminus \{۰\}‎. ‎Among other things‎, ‎the‎ ‎properties of a reduced R-module M are investigated when‎ ‎\Bbb G(M) is a bipartite graph‎.

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Paper COI Code

This Paper COI Code is JR_COMB-7-1_001. Also You can use the following address to link to this article. This link is permanent and is used as an article registration confirmation in the Civilica reference:

https://civilica.com/doc/1307322/

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If you want to refer to this Paper in your research work, you can simply use the following phrase in the resources section:
Safaeeyan, Saeed,1397,Annihilating submodule graph for modules,https://civilica.com/doc/1307322

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Type of center: دانشگاه دولتی
Paper count: 3,536
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