Quasi-socle ideals in a Gorenstein local ring

This paper explores the structure of quasi-socle ideals I=Q:m2I=Q:m2 in a Gorenstein local ring AA, where QQ is a parameter ideal and mm is the maximal ideal in AA. The purpose is to answer the problems as to when QQ is a reduction of II and when the associated graded ring G(I)=⨁n≥0In/In+1 is Cohen–Macaulay. Wild examples are explored.