Projectively equivalent ideals and Rees valuations
Rings considered are commutative, with at least two elements \(0\neq 1\). Projective equivalence of ideals was introduced by \textit{P. Samuel} [Ann. Math. (2) 56, 11--21 (1952; Zbl 0049.02301)] and his studies were continued by \textit{M. Nagata} [Mem. Coll. Sci., Univ. Kyoto, Ser. A 30, 165--175 (1957; Zbl 0095.02305)]. The authors build upon the work of \textit{S. McAdam, L. J. Ratliff jun}. and \textit{J. D. Sally} [in: Commutative algebra, Proc. Microprogram, Berkeley/CA (USA) 1989, Publ., Math. Sci. Res. Inst. 15, 391--405 (1989; Zbl 0742.13004)] who used some results of \textit{D. Rees} [J. Lond. Math. Soc. 31, 221--228 (1956; Zbl 0074.26303)]. Here is one of the authors' results: Let \(I\) and \(J\) be regular ideals of the Noetherian ring \(R\). The following are equivalent: (1) \(I\) and \(J\) are projectively equivalent, (2) The set of the Rees valuation rings for \(I\) and for \(J\) coincide and the values of \(I\) and \(J\) with respect to these Rees valuation rings are proportional. In particular, if the ideal \(I\) has only one Rees valuation ring \(V\), then the ideals \(J\) projectively equivalent to \(I\) are precisely the ideals \(J\) such that Rees\(J=\{V\}\). The strength of this paper is in its numerous examples and remarks.
scientific article; zbMATH DE number 2119558
scientific article