Quasi-socle ideals in Gorenstein numerical semigroup rings

Quasi-socle ideals, that is the ideals I of the form I=Q:mq in Gorenstein numerical semigroup rings over fields are explored, where Q is a parameter ideal, and m is the maximal ideal in the base local ring, and q⩾1 is an integer. The problems of when I is integral over Q and of when the associated graded ring G(I)=⊕n⩾0In/In+1 of I is Cohen–Macaulay are studied. The problems are rather wild; examples are given.