On \(t\)-closedness of generalized power series rings (Q5957269)

Let \(A\subset B\) be an extension of commutative rings. We say that \(A\) is \(t\)-closed in \(B\) if, whenever \(b^2-ab\), \(b^3-ab^2\in A\) for \(a\in A\) and \(b\in B\), then \(b\in A\). We say that property \({\mathcal P}_1(A,B)\) holds if, whenever \(ab\in A\) for \(a\in A\) and \(b\in B\), then \(ab^2\in A\). NEWLINENEWLINENEWLINEThe main result of this paper is that if \(A\subset B\) is an extension of commutative rings satisfying property \({\mathcal P}_1(A,B)\) and \(A\) is \(t\)-closed in \(B\), then the ring \([[A^{S,\leq}]]\) of generalized power series is \(t\)-closed in \([[B^{S, \leq}]]\), whose \((S,\leq)\) is a torsion-free cancellative ordered monoid. As a special case, this result holds for power series rings in any number of indeterminates. The case for a finite number of indeterminates was proved by \textit{A. Benhissi} [Arch. Math. 73, No. 2, 109-113 (1999; Zbl 0949.13017)].