Title: Noetherian domains with only finitely many star operations
Speaker: Evan Houston - University of North Carolina - Charlotte
Abstract. Recall that a nonzero ideal I of a domain R is said to be divisorial if (I−1)−1 = I (where I−1 = (R:I) = {x∈qf(R) | xI⊆R}). From work of Bass and Matlis, we know that a Noetherian domain R has all nonzero ideals divisorial if and only if R has Krull dimension one and M−1 is a 2-generated R-module for each maximal ideal M of R. We begin with an ideal-theoretic proof of this result. Next, recall that a star operation on a domain R is a map ∗ from the set of nonzero fractional ideals of R to itself such that, for all u ∈ qf(R) and nonzero fractional ideals I,J of R, we have (1) (aI)∗ = aI∗ and (a)∗ = (a), (2) I ⊆ I∗ and I ⊆ J implies I∗ ⊆ J∗, and (3) (I∗)∗ = I∗. Simple examples include the d-operation (Id = I for all I) and the v-operation (Iv = (I−1)−1). One sees easily that d≤∗≤v (that is, I ⊆ I∗ ⊆ Iv for all I). Hence R has all ideals divisorial if and only if R admits only one star operation. Motivated by this, we discuss our attempts to characterize Noetherian domains that admit only finitely many star operations.
