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The generalized total graph of modules respect to proper submodules over commutative rings. | ||
| Journal of Algebra and Related Topics | ||
| مقاله 3، دوره 2، شماره 1، شهریور 2014، صفحه 27-42 اصل مقاله (318.77 K) | ||
| نوع مقاله: Research Paper | ||
| نویسندگان | ||
| N. K. Tohidi* 1؛ F. Esmaeili Khalil Saraei2؛ S. A. Jalili1 | ||
| 1Islamic Azad University | ||
| 2University of Tehran | ||
| چکیده | ||
| Let $M$ be a module over a commutative ring $R$ and let $N$ be a proper submodule of $M$. The total graph of $M$ over $R$ with respect to $N$, denoted by $T(\Gamma_{N}(M))$, have been introduced and studied in [2]. In this paper, A generalization of the total graph $T(\Gamma_{N}(M))$, denoted by $T(\Gamma_{N,I}(M))$ is presented, where $I$ is an ideal of $R$. It is the graph with all elements of $M$ as vertices, and for distinct $m,n\in M$, the vertices $m$ and $n$ are adjacent if and only if $m+n\in M(N,I)$, where $M(N,I)=\{m\in M : rm\in N+IM \ for \ some \ \ r\in R-I\}$. The main purpose of this paper is to extend the definitions and properties given in [2] and [12] to a more general case. | ||
| کلیدواژهها | ||
| Total graph؛ prime submodule؛ module | ||
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