Year: 1397
Publish place:
Transactions on Combinatorics، Vol: 7، Issue: 1
COI: JR_COMB-7-1_001
Language: EnglishView: 39
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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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