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Keywords:
semiring; co-ideal; maximal co-ideal
semiring; co-ideal; maximal co-ideal
Summary:
Let $R$ be a commutative semiring with non-zero identity. In this paper, we introduce and study the graph $\Omega(R)$ whose vertices are all elements of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if the product of the co-ideals generated by $x$ and $y$ is $R$. Also, we study the interplay between the graph-theoretic properties of this graph and some algebraic properties of semirings. Finally, we present some relationships between the zero-divisor graph $\Gamma(R)$ and $\Omega(R)$.
References:
[1] Akbari S., Habibi M., Majidinya A., Manaviyat R.: A note on co-maximal graph of non-commutative rings. Algebr. Represent. Theory 16 (2013), 303–307. DOI 10.1007/s10468-011-9309-z | MR 3035995
[2] Anderson D.F., Livingston P.S.: The zero-divisor graph of a commutative ring. J. Algebra 217 (1999), 434–447. DOI 10.1006/jabr.1998.7840 | MR 1700509 | Zbl 1035.13004
[3] Beck I.: Coloring of commutative rings. J. Algebra 116 (1988), 208–226. DOI 10.1016/0021-8693(88)90202-5 | MR 0944156 | Zbl 0654.13001
[4] Chaudhari J.N., Ingale K.J.: Prime avoidance theorem for co-ideals in semirings. Research J. Pure Algebra 1(9) (2011), 213–216. MR 3020971
[5] Ebrahimi Atani S.: The zero-divisor graph with respect to ideals of a commutative semiring. Glas. Mat. 43(63) (2008), 309-320. DOI 10.3336/gm.43.2.06 | MR 2460702 | Zbl 1162.16031
[6] Ebrahimi Atani S.: An ideal-based zero-divisor graph of a commutative semiring. Glas. Mat. 44(64) (2009), 141–153. DOI 10.3336/gm.44.1.07 | MR 2525659 | Zbl 1181.16041
[7] Ebrahimi Atani S., Dolati Pish Hesari S., Khoramdel M.: Strong co-ideal theory in quotients of semirings. J. Adv. Res. Pure Math. 5 (2013), no. 3, 19–32. MR 3041341
[8] Ebrahimi Atani S., Dolati Pish Hesari S., Khoramdel M.: A fundamental theorem of co-homomorphisms for semirings. Thai J. Math. 12 (2014), no. 2, 491–497. MR 3217354 | Zbl 1310.16034
[9] Golan J.S.: Semirings and Their Applications. Kluwer Academic Publishers, Dordrecht, 1999. MR 1746739 | Zbl 0947.16034
[10] Maimani H.R., Salimi M., Sattari A., Yassemi S.: Comaximal graph of commutative rings. J. Algebra 319 (2008), 1801–1808. DOI 10.1016/j.jalgebra.2007.02.003 | MR 2383067 | Zbl 1141.13008
[11] Sharma P.K., Bhatwadekar S.M.: A note on graphical representation of rings. J. Algebra 176 (1995), 124–127. DOI 10.1006/jabr.1995.1236 | MR 1345297 | Zbl 0838.05051
[12] West D.B.: Introduction to Graph Theory. Prentice-Hall of India Pvt. Ltd, 2003. MR 1367739 | Zbl 1121.05304
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