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Annihilating submodule graph for modules | ||
| Transactions on Combinatorics | ||
| مقاله 1، دوره 7، شماره 1، خرداد 2018، صفحه 1-12 اصل مقاله (269.83 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2017.21462 | ||
| نویسنده | ||
| Saeed Safaeeyan* | ||
| Department of mathematical Sciences, Yasouj university,Yasouj, 75918-74831, IRAN. | ||
| چکیده | ||
| 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 $N*K=0$. In this paper we show that $\Bbb G(M)$ is a connected graph, ${\rm diam}(\Bbb G(M))\leq 3$, and ${\rm gr}(\Bbb G(M))\leq 4$ 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 $\Bbb A^*(M)\neq \Bbb S(M)\setminus \{0\}$ if and only if $M$ is a uniform $R$-module, ${\rm ann}(M)$ is a semi-prime ideal of $R$ and $\Bbb A^*(M)\neq \Bbb S(M)\setminus \{0\}$. 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 $\Bbb A^*(M)=\Bbb S(M)\setminus \{0\}$. Among other things, the properties of a reduced $R$-module $M$ are investigated when $\Bbb G(M)$ is a bipartite graph. | ||
| کلیدواژه‌ها | ||
| Module؛ Annihilating submodule graph؛ Complete graph | ||
| مراجع | ||
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