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

Title & Authors
TOTAL GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO IDENTITY-SUMMAND ELEMENTS
Atani, Shahabaddin Ebrahimi; Hesari, Saboura Dolati Pish; Khoramdel, Mehdi;

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($\small{{\Gamma}(R)}$), and investigate basic properties of S(R) which help us to gain interesting results about T($\small{{\Gamma}(R)}$) and its subgraphs.
Keywords
I-semiring;minimal prime co-ideal;identity-summand graph;total identity-summand graph;
Language
English
Cited by
1.
TOTAL IDENTITY-SUMMAND GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO A CO-IDEAL,;;;

대한수학회지, 2015. vol.52. 1, pp.159-176
2.
THE ANNIHILATOR IDEAL GRAPH OF A COMMUTATIVE RING,;;;;

대한수학회지, 2015. vol.52. 2, pp.417-429
1.
TOTAL IDENTITY-SUMMAND GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO A CO-IDEAL, Journal of the Korean Mathematical Society, 2015, 52, 1, 159
2.
THE ANNIHILATOR IDEAL GRAPH OF A COMMUTATIVE RING, Journal of the Korean Mathematical Society, 2015, 52, 2, 417
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.

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.

3.
D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra 320 (2008), no. 7, 2706-2719.

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.

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.

6.
D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative rings, J. Algebra 217 (1999), no. 2, 434-447.

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.

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.

10.
Z. Barati, K. Khashyarmanesh, F. Mohammadi, and K. Nafar, On the associated graphs to a commutative ring, J. Algebra Appl. 12 (2013), 1250184.

11.
I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208-226.

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.

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.

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.

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.

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.

18.
S. Ebrahimi Atani, The identity-summand graph of commutative semirings, J. Korean Math. Soc. 51 (2014), no. 1, 189-202.

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.

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.