A note on complete rings of quotients and McCoy rings

Abstract

If a (commutative unital) ring \(A\) is reduced and coincides with its total quotient ring, then \(A\) satisfies Property A (that is, \(A\) is a McCoy ring) if and only if the inclusion of \(A\) in its complete ring of quotients \(C(A)\) is a survival extension. The “if” assertion fails if one deletes the hypothesis that \(A\) is reduced. This is shown by using the idealization construction to construct a suitable ring \(A\) and then identifying its complete ring of quotients (which turns out to be a related idealization). Related characterizations of von Neumann regular rings are also given with the aid of the going-down property GD of ring extensions. For instance, a ring \(A\) is von Neumann regular if and only if \(A\) is a reduced McCoy ring that coincides with its total quotient ring such that \(A \subseteq C(A)\) satisfies GD.