Goto Numbers of a Numerical Semigroup Ring and the Gorensteiness of Associated Graded Rings

Abstract

The Goto number of a parameter ideal Q in a Noetherian local ring (R,m) is the largest integer q such that Q : m^q is integral over Q. The Goto numbers of the monomial parameter ideals of R = k[[x^{a_1}, x^{a_2},..., x_{a_{\nu}}]] are characterized using the semigroup of R. This helps in computing them for classes of numerical semigroup rings, as well as on a case-by-case basis. The minimal Goto number of R and its connection to other invariants is explored. Necessary and sufficient conditions for the associated graded rings of R and R/x^{a_1}R to be Gorenstein are also given, again using the semigroup of R.