GOTO NUMBERS OF A NUMERICAL SEMIGROUP RING AND THE GORENSTEINESS OF ASSOCIATED GRADED RINGS

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(a1),x(a2), . . . , x(av)]] 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(a1) R to be Gorenstein are also given, again using the semigroup of R.