Quasi-socle ideals in Gorenstein numerical semigroup rings

Abstract

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.

Quasi-socle ideals and Goto numbers of parameters

“…We put I = Q : m q and refer to those ideals as quasi-socle ideals in A. In this paper we are interested in the following question about quasi-socle ideals I, which are also the main subject of the researches [1][2][3]. The present research is a continuation of [1][2][3] and aims mainly at the analysis of the case where A is a complete intersection with dim A = 1.…”

“…In this paper we are interested in the following question about quasi-socle ideals I, which are also the main subject of the researches [1][2][3]. The present research is a continuation of [1][2][3] and aims mainly at the analysis of the case where A is a complete intersection with dim A = 1. Following Heinzer and Swanson [4], for each parameter ideal Q in a Noetherian local ring (A, m) we define g(Q ) = max{q ∈ Z | Q : m q ⊆ Q } and call it the Goto number of Q .…”

“…As is suggested in [1], the one-dimensional case is substantially different from higher-dimensional cases and more complicated to control. This observation has led Goto, Kimura, Matsuoka, and Phuong to the researches [2,3]. In [2] quasi-socle ideals with respect to monomial parameter ideals in Gorenstein numerical semigroup rings over fields are explored.…”

“…This observation has led Goto, Kimura, Matsuoka, and Phuong to the researches [2,3]. In [2] quasi-socle ideals with respect to monomial parameter ideals in Gorenstein numerical semigroup rings over fields are explored. In [3] the authors assumed that G(m) = n≥0 m n /m n+1 is a Gorenstein ring and parameter ideals Q are diagonal, that is, Q is generated by a system of parameters of the form x…”

“…A which generate a reduction of the maximal ideal m. The present research is a continuation of [1][2][3] and the main purpose is to go beyond the restriction in [3] that the parameter ideals Q = (x a 1 1 , x a 2 2 , . .…”