On linearly equivalent ideal topologies (Q1086633)

If \(b\) is a non-nilpotent element of a Noetherian ring \(A\), \(Z=\cup \{(0):\) \(b^ nA| n\geq 1\},\quad\phi: A\to A/Z\) is the canonical homomorphism, then \(T(bA)\) denotes the \(A\)-module \(\phi (A)_{(S)}\cap \phi (A)[1/\phi(b)]\), where \(S=\phi (A)-\cup \{P| P\in E(\phi (b)\phi (A))\}\) (here \(\phi(A)_{(S)}\) is the localization of the ring \(\phi(A)\) at the subset S; the definition of \(E(\phi(b)\phi(A))\) can be found in the paper, ''Essential sequences'', J. Algebra 95, 217-235 (1985; Zbl 0599.13004) by \textit{S. McAdam} and \textit{L. J. Ratliff} jun. If I is an ideal of a Noetherian ring A then the Rees ring R(A,I) of A with respect to I is introduced - this is the graded subring A[u,tI] of A[u,t], where t is an indeterminate and \(u=1/t\). For every \(n\geq 0\), \(I^{[n]}\) denotes the set \(u^ nT(uR(A,I))\cap A\). - It is proved that for every Noetherian ring A and every ideal I of A the filtrations, determined by the ideals \(I^{[n]}\) and \(I^ n\) respectively, are equivalent, that is there exists \(K\geq 0\) such that \(I^ m\subset I^{[m]}\subset I^{m- K}\) for all \(m\geq K.\) The following description of a Cohen-Macaulay ring is obtained: a Noetherian ring A is a Cohen-Macaulay ring if and only if A is unmixed and \(I^{[n]}=I^ n\) for all ideals of the principal class in A and for all \(n\geq 1\).