Filtrations, Rees rings, and ideal transforms (Q1078616)

Let I, J be ideals and S a multiplicatively closed set of a Noetherian ring R. \textit{P. Schenzel} [''Finiteness of relative Rees rings and asymptotic prime divisors'' (preprint)] has studied the graded extension ring \({\mathcal R}(f(I,J))\) of the Rees ring \({\mathcal R}(I)=\oplus^{\infty}_{-\infty}I^ n\quad whose\) n-th component is \(\cup_{m}(I^ n:J^ m),\quad and\) \textit{L. J. Ratliff} jun. [J. Pure Appl. Algebra 41, 67-77 (1986)] has studied \({\mathcal R}(g(I,S))\) whose n-th component is \((I^ n)_ S\). The author begins by showing that, for suitable \(z\in R\), both extensions are of the form \({\mathcal R}(f(I,Rz))={\mathcal R}(g(I,\{z^ n| n\geq 0\})\), and that they are ideal transforms of \({\mathcal R}(I)\). Properties of ideal transforms then provide him with quick proofs of criteria [due to Schenzel and Ratliff (loc. cit.)] for \({\mathcal R}(f(I,J))\) and \({\mathcal R}(g(I,S))\) to be finite/integral over R(I). The ''finite over'' case is characterized also in other ways. First, for J regular and all \(n\geq 0\), m(n) denotes the least integer m which \(\cup_{i}(I^ n:J^ i)=I^ n:J^ m,\) k(n) the least k such that \(I^ r\supseteq I^{k+r}:J^ n\) for all \(r\geq 0\) and h(n) the least k such that \(I^{k+r}:J^ n=I^ r(I^ h:J^ n)\) for all \(r\geq 0\). After establishing the existence of k(n), h(n) the author proves that \({\mathcal R}(f(I,J))\) is finite over \({\mathcal R}(I)\) if, and only if, any one of the following equivalent properties holds: \((i)\quad m(n)\) is eventually constant; \((ii)\quad k(n)\) is eventually constant; \((iii)\quad h(n)\) is bounded.