A note on complete rings of quotients and McCoy rings

摘要

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.