DOI QR코드

DOI QR Code

TOTAL GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO IDENTITY-SUMMAND ELEMENTS

  • Received : 2013.10.15
  • Published : 2014.05.01

Abstract

Let R be an I-semiring and S(R) be the set of all identity-summand elements of R. In this paper we introduce the total graph of R with respect to identity-summand elements, denoted by T(${\Gamma}(R)$), and investigate basic properties of S(R) which help us to gain interesting results about T(${\Gamma}(R)$) and its subgraphs.

Keywords

References

  1. A. Abbasi and S. Habibi, The total graph of a commutative ring with respect to proper ideals, J. Korean Math. Soc. 49 (2012), no. 1, 85-98. https://doi.org/10.4134/JKMS.2012.49.1.085
  2. S. Akbari, D. Kiani, F. Mohammadi, and S. Moradi, The total graph and regular graph of a commutative ring, J. Pure Appl. Algebra 213 (2009), no. 12, 2224-2228. https://doi.org/10.1016/j.jpaa.2009.03.013
  3. D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra 320 (2008), no. 7, 2706-2719. https://doi.org/10.1016/j.jalgebra.2008.06.028
  4. D. F. Anderson and A. Badawi, On the total graph of a commutative ring without the zeoro element, J. Algebra Appl. 11 (2012), no. 4, 1250074, 18 pp. https://doi.org/10.1142/S0219498812500740
  5. D. F. Anderson and A. Badawi, The generalized total graph of a commutative ring, J. Algebra Appl. 12 (2013), no. 5, 1250212, 18 pp. https://doi.org/10.1142/S021949881250212X
  6. D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative rings, J. Algebra 217 (1999), no. 2, 434-447. https://doi.org/10.1006/jabr.1998.7840
  7. T. Asir and T. Chelvam, The intersection graph of gamma sets in the total graph II, J. Algebra Appl. 12 (2013), no. 4, 1250199, 18 pp. https://doi.org/10.1142/S021949881250199X
  8. M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison Wesley Publishing Company, 1969.
  9. M. Axtell, J. Coykendall, and J. Stickles, Zero-divisor graphs of polynomials and power series over commutative rings, Comm. Algebra 33 (2005), no. 6, 2043-2050. https://doi.org/10.1081/AGB-200063357
  10. Z. Barati, K. Khashyarmanesh, F. Mohammadi, and K. Nafar, On the associated graphs to a commutative ring, J. Algebra Appl. 12 (2013), 1250184. https://doi.org/10.1142/S0219498812501848
  11. I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208-226. https://doi.org/10.1016/0021-8693(88)90202-5
  12. A. Bondy and U. S. R. Murty, Graph Theory, Graduate Texts in Mathematics, 244. Springer, New York, 2008.
  13. T. Chelvam and T. Asir, On the total graph and its complement of a commutative ring, Comm. Algebra 41 (2013), no. 10, 3820-3835. https://doi.org/10.1080/00927872.2012.678956
  14. T. Chelvam and T. Asir, The intersection graph of gamma sets in the total graph I, J. Algebra Appl. 12 (2013), 1250198, 18 pp. https://doi.org/10.1142/S0219498812501988
  15. S. Ebrahimi Atani, The zero-divisor graph with respect to ideals of a commutative semiring, Glas. Mat. Ser. III 43(63) (2008), no. 2, 309-320. https://doi.org/10.3336/gm.43.2.06
  16. S. Ebrahimi Atani, An ideal-based zero-divisor graph of a commutative semiring, Glas. Mat. Ser. III 44(64) (2009), no. 1, 141-153. https://doi.org/10.3336/gm.44.1.07
  17. S. Ebrahimi Atani, S. Dolati Pish Hesari, and M. Khoramdel, Strong co-ideal theory in quotients of semirings, J. Adv. Res. Pure Math. 5 (2013), no. 3, 19-32. https://doi.org/10.5373/jarpm.1482.061212
  18. S. Ebrahimi Atani, The identity-summand graph of commutative semirings, J. Korean Math. Soc. 51 (2014), no. 1, 189-202. https://doi.org/10.4134/JKMS.2014.51.1.189
  19. S. Ebrahimi Atani and F. Esmaeili Khalil Saraei, The total graph of a commutative semiring, An. Stiint. Univ. "Ovidius" Constanta Ser. Mat. 21 (2013), no. 2, 21-33.
  20. 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 (2011), no. 1, 23-34.
  21. S. Ebrahimi Atani and A. Yousefian Darani, Zero-divisor graphs with respect to primal and weakly primal ideals, J. Korean Math. Soc. 46 (2009), no. 2, 313-325. https://doi.org/10.4134/JKMS.2009.46.2.313
  22. J. S. Golan, Semirings and Their Applications, Kluwer Academic Publishers Dordrecht, 1999.
  23. J. Kist, Minimal Prime Ideals In Commutative Semigroups, Proc. Lond. Math. Soc. (3) 13 (1963), 31-50.
  24. H. Wang, On rational series and rational language, Theoret. Comput. Sci. 205 (1998), no. 1-2, 329-336.

Cited by

  1. THE ANNIHILATOR IDEAL GRAPH OF A COMMUTATIVE RING vol.52, pp.2, 2015, https://doi.org/10.4134/JKMS.2015.52.2.417
  2. TOTAL IDENTITY-SUMMAND GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO A CO-IDEAL vol.52, pp.1, 2015, https://doi.org/10.4134/JKMS.2015.52.1.159