Quasi-socle ideals and Goto numbers of parameters

Goto numbers g(Q)=max{q∈Z∣Q:mqis integral overQ} for certain parameter ideals QQ in a Noetherian local ring (A,m)(A,m) with Gorenstein associated graded ring G(m)=⨁n≥0mn/mn+1 are explored. As an application, the structure of quasi-socle ideals I=Q:mqI=Q:mq (q≥1)(q≥1) in a one-dimensional local complete intersection and the question of when the graded rings G(I)=⨁n≥0In/In+1 are Cohen–Macaulay are studied in the case where the ideals II are integral over QQ.