ON THE EXTENDED TOTAL GRAPH OF MODULES OVER COMMUTATIVE RINGS

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

References

1. D. D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra, 159(2) (1993), 500-514.
2. D. F. Anderson, M. C. Axtell and J. A. Stickles, Jr., Zero-divisor graphs in commutative rings, Commutative Algebra, Noetherian and Non-Noetherian Perspectives, eds. M. Fontana, S. E. Kabbaj, B. Olberding and I. Swanson, Springer-Verlag, New York, (2011), 23-45.
3. D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320(7) (2008), 2706-2719.
4. D. F. Anderson and A. Badawi, The generalized total graph of a commutative ring, J. Algebra Appl., 12(5) (2013), 1250212 (18 pp).
5. D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217(2) (1999), 434-447.
6. D. F. Anderson and S. B. Mulay, On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra, 210(2) (2007), 543-550.
7. I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208-226.
8. S. Ebrahimi Atani and S. Habibi, The total torsion element graph of a module over a commutative ring, An. Stiint. Univ. \Ovidius" Constanta Ser. Mat., 19(1) (2011), 23-34.

9. F. Esmaeili Khalil Saraei, The total torsion element graph without the zero el- ement of modules over commutative rings, J. Korean Math. Soc., 51(4) (2014), 721-734.
10. F. Esmaeili Khalil Saraei, H. Heydarinejad Astaneh and R. Navidinia, The total graph of a module with respect to multiplicative-prime subsets, Rom. J. Math. Comput. Sci., 4(2) (2014), 151-166.