Filtrations, Prüferian closure relative to a module (Q2785932)

The author studies some elementary properties of the integral closure of a filtration relative to a module. Let \(f=(I_n)_{n\geq 0}\) be a filtration of a commutative ring \(A\) (i.e., \(I_n\) are ideals of \(A\), \(I_0=A\), \(I_{n+1} \subset I_n\), and \(I_mI_n \subset I_{m+n}\) for any \(m,n)\) and \(M\) an \(A\)-module. An element \(x\) of \(A\) is said to be \(M\)-integral over \(f\) if \((x^m+a_1x^{m-1} +\cdots +a_m)M=0\) for some \(a_i\in I_i\). The prüferian closure \((f,M)^*= (P_n(f,M))_{n\geq 0}\) of \(f\) relative to \(M\) is defined by \(P_n(f,M)= \{x\in A\mid x\) is \(M\)-integral over \(f^{(n)}\}\), where \(f^{(n)}= (I_{nk})_{k\geq 0}\).NEWLINENEWLINENEWLINEThe author examines the behavior of the prüferian closure under localization and exact sequences of modules, and determines \((f,M)^*\) when \(A\) is a noetherian ring of dimension at most one.NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].