Formally integrally closed domains and the rings \(R((X))\) and \(R\{\{X\}\}\)
Let \(R\) be an integral domain. Denote by \(F(R)\) the set of nonzero fractional ideals of \(R\) and by \(F^{*} (R)\) the finitely generated members of \(F(R)\). If \( I \in F(R)\), \(I_{v}: = (I^{-1})^{-1}\) and \(I_{t} : = \bigcup J_{v}\), where \(J\) runs over the set \(\{ J \in F^{*} (R)| J \subseteq I \}\). Moreover, for \(f \in RX\), \(A_{f}\) stands for the ideal of \(R\) generated by the coefficients of \(f\). The ring \(R\) is said to be formally integrally closed if \((A_{fg})_{t} = (A_{f} A_{g})_{t}\) for all nonzero \(f,g \in RX\). In this paper, the authors study the rings \(R((X))= RX_{N}\) and \(R((X)) = RX_{N_{t}}\), where \(N = \{f \in RX \mid A_{f} = R \}\) and \(N_{t} = \{f \in RX \mid (A_{f})_{t} = R \}\), as a natural extension of the well-known properties of \(R[X]_{N}\) and \(R[X]_{N_{v}}\). They essentially prove the equivalence of the following statements: (i) \(R\) is a Krull domain (resp., \(R\) is a Dedekind domain). (ii) \(R((X))\) is a Krull domain (resp., \(R((X))\) is a Dedekind domain). (iii) \(R((X))\) is a Euclidean domain (resp., \(R((X))\) is a Euclidean domain). (iv) Every (principal) ideal of \(R((X))\) is extended from \(R\) (resp., every (principal) ideal of \(R((X))\) is extended from \(R\). (v) \(R\) is formally integrally closed and every prime ideal of \(R((X))\) is extended from \(R\).
scientific article; zbMATH DE number 1139782
scientific article