ON THE EXTENDED TOTAL GRAPH OF MODULES OVER COMMUTATIVE RINGS

Year 2019, , 77 - 86, 08.01.2019

F. Esmaeili Khalil Saraei E. Navidinia

https://doi.org/10.24330/ieja.504118

Abstract

 Let $M$ be a module over a commutative ring $R$ and $U$ a nonempty proper subset of $M$.

In this paper, the extended total graph, denoted by $ET_{U}(M)$, is presented, where  $U$ is a

multiplicative-prime subset of $M$. 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 $rm+sn\in U$ for some $r,s\in R\setminus (U:M)$. We also study the two (induced) subgraphs $ET_{U}(U)$ and $ET_{U}(M\setminus U)$, with vertices $U$ and $M\setminus U$, respectively. Among other things, the diameter and the girth of $ET_{U}(M)$ are also studied.

Keywords

Total graph, prime submodule, multiplicative-prime subset

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