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

https://izlik.org/JA57EH56HG

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

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• Department of Mathematics,
• The University of Tennessee,
• Knoxville, TN 37996, USA
• e-mail: [e-mail]