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

Year 2008, Volume: 4 Issue: 4, 63 - 82, 01.12.2008

John D. Lagrange

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

• S. Argyros, A decomposition of complete Boolean algebras, Pacific J. Math., 87 (1980), 1-9.
• 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.
• 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.
• D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999), 434-447.
• 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.
• I. Beck, Coloring of commutative rings, J. Algebra, 116 (1988), 208-226.
• F. DeMeyer and K. Schneider, Automorphisms and zero-divisor graphs of com- mutative rings, Internat. J. Commutative Rings, 1 (3) (2002), 93-106.
• N.J. Fine, L. Gillman, and J. Lambek, Rings of Quotients of Rings of Func- tions, McGill University Press, Montreal, 1965.
• J.A. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker, New York, 1988.
• W. Just and M. Weese, Discovering Modern Set Theory, II, American Mathe- matical Society, Providence, 1997.
• J.D. LaGrange, Complemented zero-divisor graphs and Boolean rings, J. Alge- bra, 315 (2007), 600-611.
• J. Lambek, Lectures on Rings and Modules, Blaisdell Publishing Company, Waltham, 1966.
• R. Levy and J. Shapiro, The zero-divisor graph of von Neumann regular rings, Comm. Algebra, 30 (2) (2002), 745-750.
• T.G. Lucas, The diameter of a zero divisor graph, J. Algebra, 301 (2006) 174- 193.
• J.D. Monk, Introduction to Set Theory, McGraw-Hill, New York, 1969.
• S.B. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra, 30 (7) (2002), 3533-3558.
• B. Stenstr¨om, Rings of Quotients, Springer-Verlag, New York, 1975. John D. LaGrange
• Department of Mathematics,
• The University of Tennessee,
• Knoxville, TN 37996, USA
• e-mail: [e-mail]