Projective equivalence of ideals in Noetherian integral domains

TL;DR

This paper proves the existence of finite separable extensions of Noetherian integral domains where ideals become projectively equivalent to radical ideals, with specific results for Dedekind domains.

Contribution

It establishes the existence of finite separable extensions making ideals projectively equivalent to radical ideals in Noetherian domains.

Findings

Existence of a finite separable extension domain A with all Rees integers equal to m.

Rad(IA) is a projectively full radical ideal in certain cases.

In Dedekind domains, ideals can be expressed as powers of radical ideals in finite extensions.

Abstract

Let I be a nonzero proper ideal in a Noetherian integral domain R. In this paper we establish the existence of 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. Moreover, if R has altitude one, then all the Rees integers of J = Rad(IA) are equal to one and the ideals J^m and IA have the same integral closure. Thus Rad(IA) = J is a projectively full radical ideal that is projectively equivalent to IA. In particular, if R is Dedekind, then there exists a Dedekind domain A having the following properties: (i) A is a finite separable integral extension of R; and (ii) there exists a radical ideal J of A and a positive integer m such that IA = J^m.