Power series rings over strong Mori domains.
Let \(R\) be an integral domain with quotient field \(K\). An ideal \(J\) of \(R\) is called Glaz-Vasconcelos ideal, denoted \(J\in GV(R)\), if \(J\) is finitely generated and \(J^{-1}=R\). For a fractional ideal \(I\) of \(R\), the \(w\)-closure of \(I\) is defined by \(I_w=\{x\in K;\, xJ\subseteq I\) for some \(J\in GV(R)\}\). \(I\) is called a \(w\)-ideal if \(I=I_w\). \(R\) is called a strong Mori domain if \(R\) satisfies the ascending chain condition on integral \(w\)-ideals. On the other hand, if \(\{X_{\lambda}; \lambda\in \Lambda\}\) is an arbitrary set of indeterminates over \(R\), the ring \(R[[X_{\lambda}; \lambda\in\Lambda]]_1\) is defined to be the union of the ascending net of the rings \(R[[X_{\lambda}; \lambda\in F]]\), where \(F\) runs over all finite subsets of \(\Lambda\). In this paper, the author shows: (1) if \(R\) is a noetherian domain, then \(R[[X_{\lambda}; \lambda\in\Lambda]]_1\) is a strong Mori domain; (2) if \(R\) is a strong Mori domain, then \(R[[X_{\lambda}; \lambda\in\Lambda]]_1\) is a Mori domain; but (3) there exists a strong Mori domain \(R\) such that \(RX\) is not a strong Mori domain.
scientific article; zbMATH DE number 2026933
scientific article