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ON THE EXTENDED TOTAL GRAPH OF MODULES OVER COMMUTATIVE RINGS

Year 2019, Volume: 25 Issue: 25, 77 - 86, 08.01.2019
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.

References

  • D. D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra, 159(2) (1993), 500-514.
  • 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.
  • D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320(7) (2008), 2706-2719.
  • D. F. Anderson and A. Badawi, The generalized total graph of a commutative ring, J. Algebra Appl., 12(5) (2013), 1250212 (18 pp).
  • D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217(2) (1999), 434-447.
  • 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.
  • I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208-226.
  • 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.
  • 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.
  • 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.
Year 2019, Volume: 25 Issue: 25, 77 - 86, 08.01.2019
https://doi.org/10.24330/ieja.504118

Abstract

References

  • D. D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra, 159(2) (1993), 500-514.
  • 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.
  • D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320(7) (2008), 2706-2719.
  • D. F. Anderson and A. Badawi, The generalized total graph of a commutative ring, J. Algebra Appl., 12(5) (2013), 1250212 (18 pp).
  • D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217(2) (1999), 434-447.
  • 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.
  • I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208-226.
  • 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.
  • 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.
  • 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.
There are 10 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

F. Esmaeili Khalil Saraei This is me

E. Navidinia This is me

Publication Date January 8, 2019
Published in Issue Year 2019 Volume: 25 Issue: 25

Cite

APA Saraei, F. E. K., & Navidinia, E. (2019). ON THE EXTENDED TOTAL GRAPH OF MODULES OVER COMMUTATIVE RINGS. International Electronic Journal of Algebra, 25(25), 77-86. https://doi.org/10.24330/ieja.504118
AMA Saraei FEK, Navidinia E. ON THE EXTENDED TOTAL GRAPH OF MODULES OVER COMMUTATIVE RINGS. IEJA. January 2019;25(25):77-86. doi:10.24330/ieja.504118
Chicago Saraei, F. Esmaeili Khalil, and E. Navidinia. “ON THE EXTENDED TOTAL GRAPH OF MODULES OVER COMMUTATIVE RINGS”. International Electronic Journal of Algebra 25, no. 25 (January 2019): 77-86. https://doi.org/10.24330/ieja.504118.
EndNote Saraei FEK, Navidinia E (January 1, 2019) ON THE EXTENDED TOTAL GRAPH OF MODULES OVER COMMUTATIVE RINGS. International Electronic Journal of Algebra 25 25 77–86.
IEEE F. E. K. Saraei and E. Navidinia, “ON THE EXTENDED TOTAL GRAPH OF MODULES OVER COMMUTATIVE RINGS”, IEJA, vol. 25, no. 25, pp. 77–86, 2019, doi: 10.24330/ieja.504118.
ISNAD Saraei, F. Esmaeili Khalil - Navidinia, E. “ON THE EXTENDED TOTAL GRAPH OF MODULES OVER COMMUTATIVE RINGS”. International Electronic Journal of Algebra 25/25 (January 2019), 77-86. https://doi.org/10.24330/ieja.504118.
JAMA Saraei FEK, Navidinia E. ON THE EXTENDED TOTAL GRAPH OF MODULES OVER COMMUTATIVE RINGS. IEJA. 2019;25:77–86.
MLA Saraei, F. Esmaeili Khalil and E. Navidinia. “ON THE EXTENDED TOTAL GRAPH OF MODULES OVER COMMUTATIVE RINGS”. International Electronic Journal of Algebra, vol. 25, no. 25, 2019, pp. 77-86, doi:10.24330/ieja.504118.
Vancouver Saraei FEK, Navidinia E. ON THE EXTENDED TOTAL GRAPH OF MODULES OVER COMMUTATIVE RINGS. IEJA. 2019;25(25):77-86.