Noetherian domains which admit only finitely many star operations

In this paper, we study Noetherian domains which admit only finitely many star operations. We show that such domains (which are not fields) must have Krull dimension one, and we effectively reduce to the local case. For a one-dimensional local Noetherian domain (R,M), M nonprincipal, M−1 is an overring of R, and M−1/M is naturally an R/M-vector space. We succeed in counting the number of star operations on R in several cases when .