THE IDENTITY-SUMMAND GRAPH OF COMMUTATIVE SEMIRINGS

- Journal title : Journal of the Korean Mathematical Society
- Volume 51, Issue 1, 2014, pp.189-202
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2014.51.1.189

Title & Authors

THE IDENTITY-SUMMAND GRAPH OF COMMUTATIVE SEMIRINGS

Atani, Shahabaddin Ebrahimi; Hesari, Saboura Dolati Pish; Khoramdel, Mehdi;

Atani, Shahabaddin Ebrahimi; Hesari, Saboura Dolati Pish; Khoramdel, Mehdi;

Abstract

An element r of a commutative semiring R with identity is said to be identity-summand if there exists such that r+a = 1. In this paper, we introduce and investigate the identity-summand graph of R, denoted by . 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 are studied.

Keywords

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

Language

English

Cited by

1.

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

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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.

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.

4.

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

5.

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

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.

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.

8.

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

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.

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.

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.

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.

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.

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. of Advanced Research in Pure Math. 5 (2013), no. 3, 19-32.

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.

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.