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.
Tohidi, N., Esmaeili Khalil Saraei, F., Jalili, S. (2014). The generalized total graph of modules respect to proper submodules over commutative rings.. Journal of Algebra and Related Topics, 2(1), 27-42.
MLA
N. K. Tohidi; F. Esmaeili Khalil Saraei; S. A. Jalili. "The generalized total graph of modules respect to proper submodules over commutative rings.". Journal of Algebra and Related Topics, 2, 1, 2014, 27-42.
HARVARD
Tohidi, N., Esmaeili Khalil Saraei, F., Jalili, S. (2014). 'The generalized total graph of modules respect to proper submodules over commutative rings.', Journal of Algebra and Related Topics, 2(1), pp. 27-42.
VANCOUVER
Tohidi, N., Esmaeili Khalil Saraei, F., Jalili, S. The generalized total graph of modules respect to proper submodules over commutative rings.. Journal of Algebra and Related Topics, 2014; 2(1): 27-42.