Acknowledgement
Supported by : National Natural Science Foundation of China
References
- G. Aalipour, S. Akbari, M. Behboodi, R. Nikandish, M. J. Nikmehr, and F. Shahsavari, The classification of the annihilating-ideal graphs of commutative rings, Algebra Colloq. 21 (2014), no. 2, 249-256. https://doi.org/10.1142/S1005386714000200
- G. Aalipour, S. Akbari, R. Nikandish, M. J. Nikmehr, and F. Shaveisi, On the coloring of the annihilating-ideal graph of a commutative ring, Discrete Math. 312 (2012), no. 17, 2620-2626. https://doi.org/10.1016/j.disc.2011.10.020
- G. Aalipour, S. Akbari, R. Nikandish, M. J. Nikmehr, and F. Shaveisi, Minimal prime ideals and cycles in annihilating-ideal graphs, Rocky Mountain J. Math. 43 (2013), no. 5, 1415-1425. https://doi.org/10.1216/RMJ-2013-43-5-1415
- S. Akbari and M. Mohammadian, On the zero-divisor graph of a commutative ring, J. Algebra 274 (2004), no. 2, 847-855. https://doi.org/10.1016/S0021-8693(03)00435-6
- S. Akbari and M. Mohammadian, Zero-divisor graphs of non-commutative rings, J. Algebra 269 (2006), no. 2, 462-479.
- S. Akbari and M. Mohammadian, On zero-divisor graphs of finite rings, J. Algebra 314 (2007), no. 1, 168-184. https://doi.org/10.1016/j.jalgebra.2007.02.051
- F. Aliniaeifard and M. Behboodi, Rings whose annihilating-ideal graphs have positive genus, J. Algebra Appl. 11 (2012), no. 3, 1250049, 13 pages. https://doi.org/10.1142/S0219498811005774
- F. Aliniaeifard and M. Behboodi, Commutative rings whose zero-divisor graphs have positive genus, Comm. Algebra 41 (2013), no. 10, 3629-3634. https://doi.org/10.1080/00927872.2012.673666
- D. F. Anderson, R. Levy, and J. Shapiro, Zero-divisor graphs, von Neumann regular rings, and Boolean algebras, J. Pure Appl. Algebra 180 (2003), no. 3, 221-241. https://doi.org/10.1016/S0022-4049(02)00250-5
- D. F. Anderson and P. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), no. 2, 434-447. https://doi.org/10.1006/jabr.1998.7840
- I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208-226. https://doi.org/10.1016/0021-8693(88)90202-5
- M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10 (2011), no. 4, 727-739. https://doi.org/10.1142/S0219498811004896
- M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10 (2011), no. 4, 741-753. https://doi.org/10.1142/S0219498811004902
- F. DeMeyer and L. DeMeyer, Zero-divisor graphs of semigroups, J. Algebra 283 (2005), no. 1, 190-198. https://doi.org/10.1016/j.jalgebra.2004.08.028
- F. DeMeyer, T. McKenzie, and K. Schneider, The Zero-divisor graph of a commutative semigroup, Semigroup Forum 65 (2002), no. 2, 206-214. https://doi.org/10.1007/s002330010128
- F. DeMeyer and K. Schneider, Automorphisms and zero-divisor graphs of commutative rings, Commutative rings, 25-37, Nova Sci. Publ., Hauppauge, NY, 2002.
- N. S. Karamzadeh and O. A. S. Karamzadeh, On Artinian modules over Duo rings, Comm. Algebra 38 (2010), no. 9, 3521-3531. https://doi.org/10.1080/00927870902946637
- T. Y. Lam, A First Course in Non-Commutative Rings, Springer-Verlag, New York, 1991.
- S. B. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra 30 (2002), no. 7, 3533-3558. https://doi.org/10.1081/AGB-120004502
- S. Redmond, The zero-divisor graph of a non-commutative ring, Commutative rings, 39-47, Nova Sci. Publ., Hauppauge, NY, 2002.
- S. Redmond, Structure in the zero-divisor graph of a noncommutative ring, Houston J. Math. 30 (2004), no. 2, 345-355.
- T. S. Wu, On directed zero-divisor graphs of finite rings, Discrete Math. 296 (2005), no. 1, 73-86. https://doi.org/10.1016/j.disc.2005.03.006
- T. S. Wu, Q. Liu, and L. Chen, Zero-divisor semigroups and refinements of a star graph, Discrete Math. 309 (2009), no. 8, 2510-2518. https://doi.org/10.1016/j.disc.2008.06.008
- T. S. Wu and D. C. Lu, Zero-divisor semigroups and some simple graphs, Comm. Al-gebra 34 (2006), no. 8, 3043-3052. https://doi.org/10.1080/00927870600639948
- T. S. Wu and D. C. Lu, Sub-semigroups determined by the zero-divisor graph, Discrete Math. 308 (2008), no. 22, 5122-5135. https://doi.org/10.1016/j.disc.2007.09.032