- Volume 52 Issue 1
DOI QR Code
TOTAL IDENTITY-SUMMAND GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO A CO-IDEAL
- Atani, Shahabaddin Ebrahimi (Faculty of Mathematical Sciences University of Guilan) ;
- Hesari, Saboura Dolati Pish (Faculty of Mathematical Sciences University of Guilan) ;
- Khoramdel, Mehdi (Faculty of Mathematical Sciences University of Guilan)
- Received : 2014.05.09
- Published : 2015.01.01
Let R be a semiring, I a strong co-ideal of R and S(I) the set of all elements of R which are not prime to I. In this paper we investigate some interesting properties of S(I) and introduce the total identity-summand graph of a semiring R with respect to a co-ideal I. It is the graph with all elements of R as vertices and for distinct x,
strong co-ideal;total identity-summand graph;identity-summand graph based a co-ideal;total identity-summand graph based a co-ideal
- 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
- I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208-226. https://doi.org/10.1016/0021-8693(88)90202-5
- 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
- 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
- M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison Wesley Publishing Company, 1969.
- A. Bondy and U. S. R. Murty, Graph Theory, Graduate Texts in Mathematics, 244. Springer, New York, 2008.
- D. Dolzan and P. Oblak, The zero-divisor graphs of rings and semirings, Internat. J. Algebra Comput. 22 (2012), no. 4, 1250033, 20 pp. https://doi.org/10.1142/S0218196712500336
- 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
- S. Ebrahimi Atani, An ideal-based zero-divisor graph of a commutative semiring, Glas. Matematicki 44(64) (2009), 141-153. https://doi.org/10.3336/gm.44.1.07
- 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
- 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
- S. Ebrahimi Atani, A co-ideal based identity-summand graph of a commutative semiring, submitted.
- S. Ebrahimi Atani, Total Graph of a Commutative semiring with respect to identity-summand elements, J. Korean Math. Soc. 51 (2014), no. 3, 593-607. https://doi.org/10.4134/JKMS.2014.51.3.593
- 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
- 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.
- J. S. Golan, Semirings and Their Applications, Kluwer Academic Publishers Dordrecht, 1999.
- J. Kist, Minimal prime ideals in commutative semigroups, Proc. London Math. Soc. (3) 13 (1963), 31-50.