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A GENERALIZATION OF THE ESSENTIAL GRAPH FOR MODULES OVER COMMUTATIVE RINGS

Year 2021, , 211 - 222, 05.01.2021
https://doi.org/10.24330/ieja.852234

Abstract

Let $R$ be a commutative ring with nonzero identity and let $M$ be a unitary $R$-module.
The essential graph of $M$, denoted by $EG(M)$ is a simple undirected graph
whose vertex set is $Z(M)\setminus {\rm Ann}_R(M)$ and
two distinct vertices $x$ and $y$ are adjacent if and only if ${\rm Ann}_{M}(xy)$ is an
essential submodule of $M$.
Let $r({\rm Ann}_R(M))\not={\rm Ann}_R(M)$.
It is shown that $EG(M)$ is a connected graph with
${\rm diam}(EG(M))\leq 2$.
Whenever $M$ is Noetherian, it is shown that
$EG(M)$ is a complete graph if and only if either
$Z(M)=r({\rm Ann}_R(M))$ or $EG(M)=K_{2}$ and
${\rm diam}(EG(M))= 2$ if and only if there are
$x, y\in Z(M)\setminus {\rm Ann}_R(M)$ and $\frak p\in{\rm Ass}_R(M)$ such that
$xy\not \in \frak p$. Moreover, it is proved that ${\rm gr}(EG(M))\in \{3, \infty\}$.
Furthermore, for a Noetherian module $M$ with
$r({\rm Ann}_R(M))={\rm Ann}_R(M)$ it is proved that $|{\rm Ass}_R(M)|=2$
if and only if $EG(M)$ is a complete bipartite graph that is not a star.

References

  • S. Akbari and A. Mohammadian, On the zero-divisor graph of a commutative ring, J. Algebra, 274 (2004), 847-855.
  • 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.
  • D. F. Anderson and D. Weber, The zero-divisor graph of a commutative ring without identity, Int. Electron. J. Algebra, 23 (2018), 176-202.
  • S. Babaei, Sh. Payrovi and E. Sengelen Sevim, On the annihilator submodules and the annihilator essential graph, Acta Math. Vietnam., 44 (2019), 905-914.
  • I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208-226.
  • M. Behboodi, Zero divisor graphs for modules over commutative rings, J. Commut. Algebra, 4(2) (2012), 175-197.
  • S. C. Lee and R. Varmazyar, Zero-divisor graphs of multiplication modules, Honam Math. J., 34(4) (2012), 571-584.
  • C. P. Lu, Unions of prime submodules, Houston J. Math., 23(2) (1997), 203-213.
  • M. J. Nikmehr, R. Nikandish and M. Bakhtyiari, On the essential graph of a commutative ring, J. Algebra Appl., 16(7) (2017), 1750132 (14 pp).
  • K. Nozari and Sh. Payrovi, A generalization of zero-divisor graph for modules, Publ. Inst. Math. (Beograd) (N.S.), 106(120) (2019), 39-46.
  • S. Safaeeyan, M. Baziar and E. Momtahan, A generalization of the zero-divisor graph for modules, J. Korean Math. Soc., 51(1) (2014), 87-98.
  • R. Y. Sharp, Steps in Commutative Algebra, Second edition, Cambridge University Press, Cambridge, 2000.
Year 2021, , 211 - 222, 05.01.2021
https://doi.org/10.24330/ieja.852234

Abstract

References

  • S. Akbari and A. Mohammadian, On the zero-divisor graph of a commutative ring, J. Algebra, 274 (2004), 847-855.
  • 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.
  • D. F. Anderson and D. Weber, The zero-divisor graph of a commutative ring without identity, Int. Electron. J. Algebra, 23 (2018), 176-202.
  • S. Babaei, Sh. Payrovi and E. Sengelen Sevim, On the annihilator submodules and the annihilator essential graph, Acta Math. Vietnam., 44 (2019), 905-914.
  • I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208-226.
  • M. Behboodi, Zero divisor graphs for modules over commutative rings, J. Commut. Algebra, 4(2) (2012), 175-197.
  • S. C. Lee and R. Varmazyar, Zero-divisor graphs of multiplication modules, Honam Math. J., 34(4) (2012), 571-584.
  • C. P. Lu, Unions of prime submodules, Houston J. Math., 23(2) (1997), 203-213.
  • M. J. Nikmehr, R. Nikandish and M. Bakhtyiari, On the essential graph of a commutative ring, J. Algebra Appl., 16(7) (2017), 1750132 (14 pp).
  • K. Nozari and Sh. Payrovi, A generalization of zero-divisor graph for modules, Publ. Inst. Math. (Beograd) (N.S.), 106(120) (2019), 39-46.
  • S. Safaeeyan, M. Baziar and E. Momtahan, A generalization of the zero-divisor graph for modules, J. Korean Math. Soc., 51(1) (2014), 87-98.
  • R. Y. Sharp, Steps in Commutative Algebra, Second edition, Cambridge University Press, Cambridge, 2000.
There are 13 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

F. Soheılnıa This is me

Sh. Payrovı This is me

A. Behtoeı This is me

Publication Date January 5, 2021
Published in Issue Year 2021

Cite

APA Soheılnıa, F., Payrovı, S., & Behtoeı, A. (2021). A GENERALIZATION OF THE ESSENTIAL GRAPH FOR MODULES OVER COMMUTATIVE RINGS. International Electronic Journal of Algebra, 29(29), 211-222. https://doi.org/10.24330/ieja.852234
AMA Soheılnıa F, Payrovı S, Behtoeı A. A GENERALIZATION OF THE ESSENTIAL GRAPH FOR MODULES OVER COMMUTATIVE RINGS. IEJA. January 2021;29(29):211-222. doi:10.24330/ieja.852234
Chicago Soheılnıa, F., Sh. Payrovı, and A. Behtoeı. “A GENERALIZATION OF THE ESSENTIAL GRAPH FOR MODULES OVER COMMUTATIVE RINGS”. International Electronic Journal of Algebra 29, no. 29 (January 2021): 211-22. https://doi.org/10.24330/ieja.852234.
EndNote Soheılnıa F, Payrovı S, Behtoeı A (January 1, 2021) A GENERALIZATION OF THE ESSENTIAL GRAPH FOR MODULES OVER COMMUTATIVE RINGS. International Electronic Journal of Algebra 29 29 211–222.
IEEE F. Soheılnıa, S. Payrovı, and A. Behtoeı, “A GENERALIZATION OF THE ESSENTIAL GRAPH FOR MODULES OVER COMMUTATIVE RINGS”, IEJA, vol. 29, no. 29, pp. 211–222, 2021, doi: 10.24330/ieja.852234.
ISNAD Soheılnıa, F. et al. “A GENERALIZATION OF THE ESSENTIAL GRAPH FOR MODULES OVER COMMUTATIVE RINGS”. International Electronic Journal of Algebra 29/29 (January 2021), 211-222. https://doi.org/10.24330/ieja.852234.
JAMA Soheılnıa F, Payrovı S, Behtoeı A. A GENERALIZATION OF THE ESSENTIAL GRAPH FOR MODULES OVER COMMUTATIVE RINGS. IEJA. 2021;29:211–222.
MLA Soheılnıa, F. et al. “A GENERALIZATION OF THE ESSENTIAL GRAPH FOR MODULES OVER COMMUTATIVE RINGS”. International Electronic Journal of Algebra, vol. 29, no. 29, 2021, pp. 211-22, doi:10.24330/ieja.852234.
Vancouver Soheılnıa F, Payrovı S, Behtoeı A. A GENERALIZATION OF THE ESSENTIAL GRAPH FOR MODULES OVER COMMUTATIVE RINGS. IEJA. 2021;29(29):211-22.