Quasi-socle ideals, that is the ideals I of the form I = Q : m q in a Noetherian local ring (A, m) with the Gorenstein tangent cone G(m) = n≥0 m n /m n+1 are explored, where q ≥ 1 is an integer and Q is a parameter ideal of A generated by monomials of a system x 1 , x 2 , · · · , x d of elements in A such that (x 1 , x 2 , · · · , x d ) is a reduction of m. The questions of when I is integral over Q and of when the graded rings G(I) = n≥0 I n /I n+1 and F(I) = n≥0 I n /mI n are Cohen-Macaulay are answered. Criteria for G(I) and R(I) = n≥0 I n to be Gorenstein rings are given.

Quasi-socle ideals, that is the ideals I of the form I = Q : m q 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 0 I n /I n+1 of I is Cohen-Macaulay are studied. The problems are rather wild; examples are given.

This paper is based on the translation of the Japanese version with some slight modifications.

a b s t r a c tGoto numbers g(Q ) = max{q ∈ Z | Q : m q is integral over Q } for certain parameter ideals Q in a Noetherian local ring (A, m) with Gorenstein associated graded ring G(m) = n≥0 m n /m n+1 are explored. As an application, the structure of quasi-socle ideals I = Q : m q (q ≥ 1) in a one-dimensional local complete intersection and the question of when the graded rings G(I) = n≥0 I n /I n+1 are Cohen-Macaulay are studied in the case where the ideals I are integral over Q .

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