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

Authors : John D LAGRANGE

Pages : 63-82

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Publication Date : 2008-12-01

Article Type : Research Paper

Abstract :One defines an equivalence relation on a commutative ring R by declaring elements r1, r2 ∈ R to be equivalent if and only if annRinsert ignore into journalissuearticles values(r1); = annRinsert ignore into journalissuearticles values(r2);. If [r]R denotes the equivalence class of an element r ∈ R, then it is known that |[r]R| = |[r/1]T insert ignore into journalissuearticles values(R); |, where Tinsert ignore into journalissuearticles values(R); denotes the total quotient ring of R. In this paper, we investigate the extent to which a similar equality will hold when Tinsert ignore into journalissuearticles values(R); is replaced by Qinsert ignore into journalissuearticles values(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