Two properties of the power series ring

Abstract

For a commutative ring with unity,A, it is proved that the power series ringA〚X〛is a PF-ring if and only if for any two countable subsetsSandTofAsuch thatS⫅annA(T), there existsc∈annA(T)such thatbc=bfor allb∈S. Also it is proved that a power series ringA〚X〛is a PP-ring if and only ifAis a PP-ring in which every increasing chain of idempotents inAhas a supremum which is an idempotent.