Two properties of the power series ring
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) \(AX\) 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 \(a\leq b\leftrightarrow a^ 2=ab\).
scientific article; zbMATH DE number 4148252