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.