THE CARDINALITY OF AN ANNIHILATOR CLASS IN A VON NEUMANN REGULAR RING

Abstract

One defines an equivalence relation on a commutative ring R by declaring elements r1, r2 ∈ R to be equivalent if and only if annR(r1) = annR(r2). If [r]R denotes the equivalence class of an element r ∈ R, then it is known that |[r]R| = |[r/1]T (R) |, where T(R) denotes the total quotient ring of R. In this paper, we investigate the extent to which a similar equality will hold when T(R) is replaced by Q(R), the complete ring of quotients of R. The results are applied to compare the zero-divisor graph of a reduced commutative ring to that of its complete ring of quotients.

Keywords

• von Neumann regular ring, zero-divisor graph, complete ring of quotients

References

1. S. Argyros, A decomposition of complete Boolean algebras, Pacific J. Math., 87 (1980), 1-9.
2. D.F. Anderson, A. Frazier, A. Lauve, and P.S. Livingston, The zero-divisor graph of a commutative ring, II, Lecture Notes in Pure and Applied Mathe- matics, (editors Daniel D. Anderson and Ira J. Papick), Marcel Dekker (New York, 2001), 220, pp. 61-72.
3. D.F. Anderson, R. Levy, and J. Shapiro, Zero-divisor graphs, von Neumann regular rings, and Boolean algebras, J. Pure Appl. Algebra, 180 (2003), 221- 241.
4. D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999), 434-447.
5. D.F. Anderson and S.B. Mulay, On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra, 210 (2007), 543-550.
6. I. Beck, Coloring of commutative rings, J. Algebra, 116 (1988), 208-226.
7. F. DeMeyer and K. Schneider, Automorphisms and zero-divisor graphs of com- mutative rings, Internat. J. Commutative Rings, 1 (3) (2002), 93-106.
8. N.J. Fine, L. Gillman, and J. Lambek, Rings of Quotients of Rings of Func- tions, McGill University Press, Montreal, 1965.

9. J.A. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker, New York, 1988.
10. W. Just and M. Weese, Discovering Modern Set Theory, II, American Mathe- matical Society, Providence, 1997.
11. J.D. LaGrange, Complemented zero-divisor graphs and Boolean rings, J. Alge- bra, 315 (2007), 600-611.
12. J. Lambek, Lectures on Rings and Modules, Blaisdell Publishing Company, Waltham, 1966.
13. R. Levy and J. Shapiro, The zero-divisor graph of von Neumann regular rings, Comm. Algebra, 30 (2) (2002), 745-750.
14. T.G. Lucas, The diameter of a zero divisor graph, J. Algebra, 301 (2006) 174- 193.
15. J.D. Monk, Introduction to Set Theory, McGraw-Hill, New York, 1969.
16. S.B. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra, 30 (7) (2002), 3533-3558.
17. B. Stenstr¨om, Rings of Quotients, Springer-Verlag, New York, 1975. John D. LaGrange
18. Department of Mathematics,
19. The University of Tennessee,
20. Knoxville, TN 37996, USA
21. e-mail: [e-mail]