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Year 2018, Volume: 23 Issue: 23, 157 - 166, 11.01.2018
https://doi.org/10.24330/ieja.373659

Abstract

References

  • S. Akbari and A. Mohammadian, On the zero-divisor graph of a commutative ring, J. Algebra, 274(2) (2004), 847-855.
  • D. F. Anderson and A. Badawi, On the zero-divisor graph of a ring, Comm. Algebra, 36(8) (2008), 3073-3092.
  • D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320(7) (2008), 2706-2719.
  • D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217(2) (1999), 434-447.
  • M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Alge- bra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.
  • I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208-226. [7] N. Bloom eld and C. Wickham, Local rings with genus two zero divisor graph, Comm. Algebra, 38(8) (2010), 2965-2980.
  • J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier Publishing Co., Inc., New York, 1976 I. Chakrabarty, S. Ghosh, T. K. Mukherjee and M. K. Sen, Intersection graphs of ideals of rings, Discrete Math., 309(17) (2009), 5381-5392.
  • A. M. Dhorajia, Total graph of the ring Zn  Zm, Discrete Math. Algorithms Appl., 7(1) (2015), 1550004 (9 pp).
  • A. M. Dhorajia and J. M. Morzaria, Domination in total graphs of small rings, Discrete Math. Algorithms Appl., 8(4) (2016), 1650069 (11 pp).
  • H. R. Maimani, M. R. Pournaki and S. Yassemi, Zero-divisor graph with respect to an ideal, Comm. Algebra, 34(3) (2006), 923-929.
  • H. R. Maimani, C. Wickham and S. Yassemi, Rings whose total graphs have genus at most one, Rocky Mountain J. Math., 42(5) (2012), 1551-1560.
  • M. J. Nikmehr and F. Shaveisi, The regular digraph of ideals of a commutative ring, Acta Math. Hungar., 134(4) (2012), 516-528.
  • P. K. Sharma and S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra, 176(1) (1995), 124-127.
  • S. Spiro and C. Wickham, A zero divisor graph determined by equivalence classes of zero divisors, Comm. Algebra, 39(7) (2011), 2338-2348.
  • C. Wickham, Classi cation of rings with genus one zero-divisor graphs, Comm. Algebra, 36(2) (2008), 325-345.
  • M. Ye and T. Wu, Co-maximal ideal graphs of commutative rings, J. Algebra Appl., 11(6) (2012), 1250114 (14 pp).
  • M. Ye, T. Wu, Q. Liu and J. Guo, Graph properties of co-maximal ideal graphs of commutative rings, J. Algebra Appl., 14(3) (2015), 1550027 (13 pp).

Some graph on prime ideals of a commutative ring

Year 2018, Volume: 23 Issue: 23, 157 - 166, 11.01.2018
https://doi.org/10.24330/ieja.373659

Abstract

Let R be a commutative ring with an identity. Let Spec(R) be
the set of all prime ideals of R and Max(R) be the set of all maximal ideals
of R. Let S  Max(R). We de ne an S-proper ideal sum graph on Spec(R),
denoted by 􀀀S(Spec(R); S), as an undirected graph whose vertex set is the set
Spec(R) and, for two distinct vertices P and Q, there is an arc from P to Q,
whenever P +Q M, for some maximal idealMin S. In this paper, we prove
that the complement graph of a proper sum graph 􀀀(Spec(R); S) is complete
if and only if R is an Artinian ring. We also study some basic properties of
the graph 􀀀S(Spec(R); S) such as connectivity, girth and clique number. We
explore the in
uence of the ring theoretic properties of a commutative ring R
on the proper sum graph of R and vice versa.

References

  • S. Akbari and A. Mohammadian, On the zero-divisor graph of a commutative ring, J. Algebra, 274(2) (2004), 847-855.
  • D. F. Anderson and A. Badawi, On the zero-divisor graph of a ring, Comm. Algebra, 36(8) (2008), 3073-3092.
  • D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320(7) (2008), 2706-2719.
  • D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217(2) (1999), 434-447.
  • M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Alge- bra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.
  • I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208-226. [7] N. Bloom eld and C. Wickham, Local rings with genus two zero divisor graph, Comm. Algebra, 38(8) (2010), 2965-2980.
  • J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier Publishing Co., Inc., New York, 1976 I. Chakrabarty, S. Ghosh, T. K. Mukherjee and M. K. Sen, Intersection graphs of ideals of rings, Discrete Math., 309(17) (2009), 5381-5392.
  • A. M. Dhorajia, Total graph of the ring Zn  Zm, Discrete Math. Algorithms Appl., 7(1) (2015), 1550004 (9 pp).
  • A. M. Dhorajia and J. M. Morzaria, Domination in total graphs of small rings, Discrete Math. Algorithms Appl., 8(4) (2016), 1650069 (11 pp).
  • H. R. Maimani, M. R. Pournaki and S. Yassemi, Zero-divisor graph with respect to an ideal, Comm. Algebra, 34(3) (2006), 923-929.
  • H. R. Maimani, C. Wickham and S. Yassemi, Rings whose total graphs have genus at most one, Rocky Mountain J. Math., 42(5) (2012), 1551-1560.
  • M. J. Nikmehr and F. Shaveisi, The regular digraph of ideals of a commutative ring, Acta Math. Hungar., 134(4) (2012), 516-528.
  • P. K. Sharma and S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra, 176(1) (1995), 124-127.
  • S. Spiro and C. Wickham, A zero divisor graph determined by equivalence classes of zero divisors, Comm. Algebra, 39(7) (2011), 2338-2348.
  • C. Wickham, Classi cation of rings with genus one zero-divisor graphs, Comm. Algebra, 36(2) (2008), 325-345.
  • M. Ye and T. Wu, Co-maximal ideal graphs of commutative rings, J. Algebra Appl., 11(6) (2012), 1250114 (14 pp).
  • M. Ye, T. Wu, Q. Liu and J. Guo, Graph properties of co-maximal ideal graphs of commutative rings, J. Algebra Appl., 14(3) (2015), 1550027 (13 pp).
There are 17 citations in total.

Details

Journal Section Articles
Authors

Alpesh M. Dhorajia This is me

Publication Date January 11, 2018
Published in Issue Year 2018 Volume: 23 Issue: 23

Cite

APA Dhorajia, A. M. (2018). Some graph on prime ideals of a commutative ring. International Electronic Journal of Algebra, 23(23), 157-166. https://doi.org/10.24330/ieja.373659
AMA Dhorajia AM. Some graph on prime ideals of a commutative ring. IEJA. January 2018;23(23):157-166. doi:10.24330/ieja.373659
Chicago Dhorajia, Alpesh M. “Some Graph on Prime Ideals of a Commutative Ring”. International Electronic Journal of Algebra 23, no. 23 (January 2018): 157-66. https://doi.org/10.24330/ieja.373659.
EndNote Dhorajia AM (January 1, 2018) Some graph on prime ideals of a commutative ring. International Electronic Journal of Algebra 23 23 157–166.
IEEE A. M. Dhorajia, “Some graph on prime ideals of a commutative ring”, IEJA, vol. 23, no. 23, pp. 157–166, 2018, doi: 10.24330/ieja.373659.
ISNAD Dhorajia, Alpesh M. “Some Graph on Prime Ideals of a Commutative Ring”. International Electronic Journal of Algebra 23/23 (January 2018), 157-166. https://doi.org/10.24330/ieja.373659.
JAMA Dhorajia AM. Some graph on prime ideals of a commutative ring. IEJA. 2018;23:157–166.
MLA Dhorajia, Alpesh M. “Some Graph on Prime Ideals of a Commutative Ring”. International Electronic Journal of Algebra, vol. 23, no. 23, 2018, pp. 157-66, doi:10.24330/ieja.373659.
Vancouver Dhorajia AM. Some graph on prime ideals of a commutative ring. IEJA. 2018;23(23):157-66.

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