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 cal ls an M-primary ideal I maximally generated if mu(I) = l(R/(r)), or wh ich is the same, if IM = rI for some r epsilon M, and he also proves t hat if there is a maximally generated ideal in R then there is a uniqu e biggest one (see [4]). In this paper each ring (R, M) is a local one -dimensional Cohen-Macaulay ring. Let Q be the total ring of fractions of R, and let B(M) be the ring obtained by blowing up M, i.e. B(M) = boolean OR(i greater than or equal to 1)(M(i) : M(i))(Q). We prove in Theorem 1 that if there are maximally generated ideals in R then they are the M-primary ideals of R which are ideals of B(M) too. And the bi ggest maximally generated ideal I of R is the conductor of R in B(M), i.e. (R : B(M))(R). We give in Theorem 3 an algorithm for finding I wh en the integral closure of R is a local domain with the same residue f ield as R. In section 3 there are applications to semigroup rings. We prove that I is generated by monomials in Proposition 7, and therefore semigroups are considered in the continuation. Let delta be the reduc tion exponent of M, i.e. delta = min{i:l(M(i)/M(i+1)) = e(M)} where e( M) denotes the multiplicity of M. In Proposition 10, delta is determin ed, and there is also given a sufficient condition for I not to be a p ower of M. In Propositions 11 and 12, I is determined for two special cases of semigroup rings where I is a power of M.