\(t\)-closed rings of formal power series (Q1305352)

Let \(A \subset B\) be rings. \(A\) is said to be \(t\)-closed in \(B\) if for each \(a \in A\) and \(b \in B\) such that \(b^{2}-ab, b^{3}-ab^{2} \in A\) then \(b \in A\). In this note, the author proves that if \(A\) and \(B\) satisfy that for each \(a \in A\) and \(b \in B\) such that \(ab \in A\) then \(ab^{2} \in A\) and that \(A\) is \(t\)-closed in \(B\), then the power series ring in \(n\) indeterminates \(A[[X]]\) is \(t\)-closed in \(B[[X]]\). Moreover, the author shows that these conditions are not necessary. However, some necessary and sufficient conditions for \(A[[X]]\) to be \(t\)-closed in \(B[[X]]\) are given, for example it holds when \(A \subset B\) is an integral extension and \(A\) is \(t\)-closed in \(B\).