Projective equivalence of ideals in noetherian integral domains (Q952545)

Let \(R\) be a noetherian integral domain. This paper investigates the finite integral extensions of \(R\) and the projective equivalence finite integral extensions. Especially, Theorem 1.5, shows that for a nonzero proper ideal of \(R\), there exists a finite separable integral extension domain \(A\) of \(R\) and a positive integer \(m\) such that all the Rees integers of \(IA\) are equal to \(m\) and shows that if \(R\) has altitude one, then there exists a finite integral extensions domain \(A\) of \(R\) such that \(P(IA)\) contains an ideal \(H\) whose Rees integers are all equal to one. Therefore \(H=\text{Rad}(IA)\) is a projectively full radical ideal that is projectively equivalent to \(IA\), where \(P(IA)\) is the set of integrally closed ideals projectively equivalent to \(I\). A particular case of this result where \(R\) is a Dedekind domain. Then there exists a Dedekind domain \(E\) having the following properties: (1) \(E\) is a finite separable integral extension of \(R\); and (2) there exists a radical ideal \(J\) of \(E\) and a positive integer \(m\) such that \(IE=J^{m}\). Therefore \(J\) is a projectively full radical ideal. That is projectively equivalent to \(IE\), and the Rees integers of \(J\) are all equal to one.The extension also has the property that for each maximal \(N\) of \(E\) with \(I\subseteq N\), the canonical inclusion \(\frac{R}{N\cap R}\hookrightarrow \frac{E}{N}\) is an isomorphism and \(m\) is a multiple of \([E_{(0)}:R_{(0)}]\). The main result of section 4, is: let \(I_{1}\;, \dots,\;I_{h}\) be nonzero proper ideals in a noetherian domain \(R\) and for \(i=1,\;dots,\;h\), let \(e_{i,1},\dots,\;e_{i,n_{i}}\) be the Rees integers of \(I_{i}\) and \(m_{i}=e_{i,1}.\dots e_{i,n_{i}}\). Assume that \(\text{Rees} I_{i} \cap \text{Rees} I_{j}=\emptyset\) for \(i\neq j\) in \(\{1,\dots,h\}\). Then there exists a simple free separable integral extension domain \(A\) of \(R\) such that, for \(i=1,\dots,h\), the Rees integers of \(I_{i}A\) are all equal to \(m_{i}\).