THE IDENTITY-SUMMAND GRAPH OF COMMUTATIVE SEMIRINGS

• Atani, Shahabaddin Ebrahimi ;
• Hesari, Saboura Dolati Pish ;
• Khoramdel, Mehdi
• Received : 2013.05.07
• Published : 2014.01.01
• 66 6

Abstract

An element r of a commutative semiring R with identity is said to be identity-summand if there exists $1{\neq}a{\in}R$ such that r+a = 1. In this paper, we introduce and investigate the identity-summand graph of R, denoted by ${\Gamma}(R)$. It is the (undirected) graph whose vertices are the non-identity identity-summands of R with two distinct vertices joint by an edge when the sum of the vertices is 1. The basic properties and possible structures of the graph ${\Gamma}(R)$ are studied.

Keywords

I-semiring;co-ideal;Q-strong co-ideal;co-semidomain;identity-summand graph;identity-summand element

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. S. Akbari, H. R. Maimani, and S. Yassemi, When a zero-divisor graph is planar or a complete r-partite graph, J. Algebra 270 (2003), no. 1, 169-180. https://doi.org/10.1016/S0021-8693(03)00370-3
4. 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
5. 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
6. D. F. Anderson and A. Badawi, The total graph of a commutative ring without the zero element, J. Algebra Appl. 11 (2012), no. 4, 1250074, 18 pp. https://doi.org/10.1142/S0219498812500740
7. 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
8. 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
9. D. F. Anderson and S. B. Mulay, On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra 210 (2007), no. 2, 543-550. https://doi.org/10.1016/j.jpaa.2006.10.007
10. Z. Barati, K. Khashyarmanesh, F. Mohammadi, and K. Nafar, On the associated graphs to a commutative ring, J. Algebra Appl. 11 (2012), no. 2, 1250037, 17 pp. https://doi.org/10.1142/S0219498811005610
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 with Applications, American Elsevier, New York, 1976.
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), no. 4, 1250198, 18pp. https://doi.org/10.1142/S0219498812501988
15. T. Chelvam and T. Asir, The intersection graph of gamma sets in the total graph II, J. Algebra Appl. 12 (2013), no. 4, 1250199, 14 pp. https://doi.org/10.1142/S021949881250199X
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. of Advanced Research in Pure Math. 5 (2013), no. 3, 19-32. https://doi.org/10.5373/jarpm.1482.061212
18. 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
19. J. S. Golan, Semirings and Their Applications, Kluwer Academic Publishers Dordrecht, 1999.
20. H. R. Maimani, M. R. Pournaki, and S. Yassemi, Zero-divisor graph with respect to an ideal, Comm. Algebra 34 (2006), no. 3, 923-929. https://doi.org/10.1080/00927870500441858
21. S. Spiroff and C. Wickham, A zero divisor graph determined by equivalence classes of zero divisors, Comm. Algebra 39 (2011), no. 7, 2338-2348. https://doi.org/10.1080/00927872.2010.488675
22. H. Wang, On rational series and rational language, Theoret. Comput. Sci. 205 (1998), no. 1-2, 329-336. https://doi.org/10.1016/S0304-3975(98)00103-0

Cited by

1. Semisimple semirings with respect to co-ideals theory 2017, https://doi.org/10.1142/S1793557118500699
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
3. TOTAL GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO IDENTITY-SUMMAND ELEMENTS vol.51, pp.3, 2014, https://doi.org/10.4134/JKMS.2014.51.3.593