Kerstin Pettersson

On the biggest maximally generated ideal as the conductor in the blowing up ring

Let (R, M) be a Noetherian one-dimensional local ring.C Gottlieb calls anM-primary idealI maximally generated ifμ(I)=ℓ(R/(r)), or which is the same, ifIM=rI for somer∈M, and he also proves that if there is a maximally generated ideal inR then there is a unique biggest one (see [4]). In this paper each ring (R, M) is a local one-dimensional Cohen-Macaulay ring. LetQ be the total ring of fractions ofR, and letB(M) be the ring obtained by blowing upM, i.e.B (M)=Ui≥1 (Mi:Mi)Q. We prove in Theorem 1 that if there are maximally generated ideals inR then they are theM-primary ideals ofR which are ideals ofB(M) too. And the biggest maximally generated idealÎ ofR is the conductor ofR inB(M), i.e.(R∶B(M))R. We give in Theorem 3 an algorithm for findingÎ when the integral closure ofR is a local domain with the same residue field asR. In section 3 there are applications to semigroup rings. We prove thatÎ is generated by monomials in Proposition 7, and therefore semigroups are considered in the continuation. Let σ be the reduction exponent ofM, i.e. δ=min{i∶ℓ(Mi/Mi+1) =e(M)} wheree(M) denotes the multiplicity ofM. In Proposition 10, δ is determined, and there is also given a sufficient condition forÎ not to be a power ofM. In Propositions 11 and 12Î is determined for two special cases of semigroup rings whereÎ is a power ofM.