On the relevant transform and the relevant component of an ideal (Q1094461)

If \(I\) is an ideal in a noetherian ring \(R\), then the relevant component of \(I\) is the ideal \(I^*=\bigcup (I^{i+1}:I^ i)\). Let \(R=R[tI,u]\) (where \(t\) is an indeterminate and \(u=t^{-1})\) and let \(B=R_ S\cap R[u,t]\), where \(S\) is the set of regular elements in \(R-\cup \{P\in Ass(R/uR);\quad tI\not\subseteq P\}.\) Then \(B\) is called the relevant transform of \(I\). A number of properties of these two concepts are derived and connections between them are studied. Applications are given, e.g. that the sets \(Ass(R/I^ n)^*\) and \(Ass(I^ n)^*/(I^{n+1})^*\) are monotonically increasing and eventually stable.