When are trace ideals finite?

Abstract

In this paper, we study Noetherian local rings having finite trace ideals. We proved that such rings are of dimension at most one. For the onedimensional case, we can reduce the situation that rings are Cohen-Macaulay. Then, we give a necessary condition to have finite trace ideals in terms of the value set obtained by the canonical module. We also gave the correspondence between trace ideals of R and that of the endomorphism algebra of the maximal ideal of R when R has minimal multiplicity.

The set of trace ideals of curve singularities

“…Let R be a fiber product k[[t 5 , t 8 , t 12 Then, it is clear that R is not a domain. On the other hand, since both T(k[[t 5 , t 8 , t 12 , t 14 ]]) and T(k[[t 9 , t 12 , t 16 , t 19 , t 20 , t 22 , t 23 , t 26 ]]) are finite by Example 4.3, T(R) is also finite.…”

“…As a particular question, the following is raised naturally and explored in several papers: Question 1.1. ([7, Question 3.7], [6,Question 7.16(1)], [15,19]) When does a Noetherian local ring have a finite number of trace ideals?…”

“…In [19], the second author proved that if a local domain R has a finite number of trace ideals, then dim R ≤ 1 and R is analytically unramified ([19, Lemma 2.4 and Theorem 2.6]). In the case of dimension one, it is also proven that analytically irreducible Arf local domains have a finite number of trace ideals ([19, Corollary 5.5]).…”