Two properties of the power series ring

Abstract

For a commutative ring with unity, A, it is proved that the power series ring A〚X〛 is a PF-ring if and only if for any two countable subsets S and T of A such that S⫅annA(T), there exists c∈annA(T) such that bc=b for all b∈S. Also it is proved that a power series ring A〚X〛 is a PP-ring if and only if A is a PP-ring in which every increasing chain of idempotents in A has a supremum which is an idempotent.