Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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THE GROUP OF GRAPH AUTOMORPHISMS OVER A MATRIX RING

  • Received : 2009.09.25
  • Published : 2011.03.01
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Abstract

Let R = $Mat_2(F)$ be the ring of all 2 by 2 matrices over a finite field F, X the set of all nonzero, nonunits of R and G the group of all units of R. After investigating some properties of orbits under the left (and right) regular action on X by G, we show that the graph automorphisms group of $\Gamma(R)$ (the zero-divisor graph of R) is isomorphic to the symmetric group $S_{|F|+1}$ of degree |F|+1.

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References

  1. S. Akbari and A. 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
  2. D. F. Anderson, A. Frazier, A. Lauve, and P. S. Livingston, The zero-divisor graph of a commutative ring. II, Ideal theoretic methods in commutative algebra (Columbia, MO, 1999), 61-72, Lecture Notes in Pure and Appl. Math., 220, Dekker, New York, 2001.
  3. D. F. Anderson and P. S. 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
  4. I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208-226. https://doi.org/10.1016/0021-8693(88)90202-5
  5. 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
  6. N. J. Fine and I. N. Herstein, The probability that a matrix be nilpotent, Illinois J. Math. 2 (1958), 499-504.
  7. J. Han, Half-transitive group actions in a left Artinian ring, Kyungpook Math. J. 37 (1997), no. 2, 297-303.
  8. J. Han, The zero-divisor graph under group actions in a noncommutative ring, J. Korean Math. Soc. 45 (2008), no. 6, 1647-1659. https://doi.org/10.4134/JKMS.2008.45.6.1647
  9. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer- Verlag, New York, 1991.
  10. S. P. Redmond, The zero-divisor graph of non-commutative ring, Internat. J. Commutative Rings 1 (2002), no. 4, 2003-211.
  11. S. P. Redmond, Structure in the zero-divisor graph of a noncommutative ring, Houston J. Math. 30 (2004), no. 2, 345-355.
  12. T. Wu, On directed zero-divisor graphs of nite rings, Discrete Math. 296 (2005), no. 1, 73-86. https://doi.org/10.1016/j.disc.2005.03.006

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