Pseudo-Dedekind domains and divisorial ideals in \(R[X]_ T\)
The authors introduce a notion of pseudo-Dedekind domain as a domain \(R\) in which every divisorial ideal is invertible. This generalizes the notion of pseudo-principal domain introduced by \textit{Bourbaki} and the notion of G-GCD domain (for which finite type divisorial ideals are invertible). The authors give several alternative characterizations of pseudo-Dedekind domains, for example, for an integral domain R the following conditions are equivalent: (1) \(R\) is pseudo-Dedekind, (2) \((A.B)^{-1}=A^{-1}.B^{-1}\) for all nonzero fractional ideals of \(R\), (3) \(R\) is completely integrally closed and the product of divisorial ideals is divisorial. Further, using some results which give cases when the extension or contraction of divisorial ideals remain divisorial they prove that the following statements are equivalent: (1) \(R\) is pseudo-Dedekind, (2) \(R[X]\) is pseudo-Dedekind, (3) \(R(X)\) is pseudo-Dedekind, (4) \(R(X)\) is pseudo-principal. Moreover, a group ring \(R[X,G]\) is pseudo-Dedekind if and only if \(R\) is pseudo-Dedekind and \(G\) has type \((0,0,...)\).
scientific article; zbMATH DE number 4144103
scientific article