A note on powers of ideals (Q1380040)

First, the author establishes the following result: If \(R\) is a Noetherian local ring and \(I\) is an ideal of \(R\), then \(I^n\) can be generated by a quadratic sequence for all \(n \gg 0\). Secondly, using this result the author gives proofs of the known results given below: Result 1. Let \(R\) be a Noetherian ring and \(I\) be an ideal of \(R\). Then reltype\((I^n) \leq 2\) for all \(n\gg 0\). -- [This result is found in a preprint by \textit{Johnson} and \textit{Katz}: ``Reduction numbers and relation types for higher powers of ideals in a Noetherian ring'' (1993)]. Result 2. Let \(R\) be a Noetherian ring and \(I\) be an ideal of \(R\). Then for sufficiently large \(n\), \(\text{Ass} (R/I^n)\) is independent of \(n\). In particular, \(\bigcup \{\text{Ass} (R/I^n): n\geq 1\}\) is finite. [This is in a paper by \textit{M. Brodmann}: ``Asymptotic stability of \(\text{Ass} (M/I^nM)\)'', Proc. Am. Math. Soc. 14, 16-18 (1979; Zbl 0372.13010)].