Two properties of the power series ring (Q913870)

A commutative ring with identity A is called a PF [resp. PP] ring if every principal ideal is flat [resp. principal]. The author proves the following: (1) 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 \(S\subset ann_ A(T)\), there exists \(c\in ann_ A(T)\) such that \(bc=b\) for all \(b\in S\). (2) \(A[[X]]\) 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. In (2) the supremum is relative to the partial order given by \(a\leq b\leftrightarrow a^ 2=ab\).