[Paper Review] Rings whose ideals are isomorphic to trace ideals

Toshinori Kobayashi, Ryo Takahashi|arXiv (Cornell University)|Jul 1, 2018

Commutative Algebra and Its Applications15 references4 citations

TL;DR

This paper provides a complete classification of commutative noetherian local rings in which every ideal is isomorphic to a trace ideal. It establishes that such rings are precisely artinian Gorenstein rings, 1-dimensional hypersurfaces of multiplicity at most 2, or unique factorization domains—resolving a question posed by Lindo and Pande. The results are derived through duality theory, trace ideal properties, and depth arguments in local rings of various dimensions.

ABSTRACT

Let R be a commutative noetherian ring. Lindo and Pande have recently posed the question asking when every ideal of R is isomorphic to some trace ideal of R. This paper studies this question and gives several answers. In particular, a complete answer is given in the case where R is local: it is proved in this paper that every ideal of R is isomorphic to a trace ideal if and only if R is an artinian Gorenstein ring, or a 1-dimensional hypersurface with multiplicity at most 2, or a unique factorization domain.

Motivation & Objective

• To resolve a question posed by Lindo and Pande on when every ideal of a commutative noetherian ring is isomorphic to a trace ideal.
• To characterize local rings satisfying the Lindo–Pande condition, particularly in low-depth and higher-dimensional settings.
• To clarify the relationship between trace ideals, factoriality, and Gorenstein/hypersurface singularities in local rings.
• To establish that the Lindo–Pande condition does not necessarily ascend under completion, especially in depth ≥2 rings.
• To provide a complete characterization of local rings where all ideals are isomorphic to trace ideals, unifying cases across dimension and singularity type.

Proposed method

• Using duality theory and the $R$-dual functor $(-)^*$, the paper analyzes trace ideals via the map $\lambda_M^R: M^* \otimes_R M \to R$, whose image is $\operatorname{tr}_R M$.
• Applying the trace ideal construction to ideals and modules, the paper investigates when ideals are isomorphic to trace ideals using colons and annihilators in the total quotient ring $Q$.
• Employing depth arguments and the $\mathrm{(S_2)}$ condition, the paper analyzes the structure of ideals in rings of depth ≥2, particularly focusing on height-1 and height-0 prime ideals.
• Using Krull’s intersection theorem and exact sequences involving quotient modules, the paper derives contradictions when assumptions about ideal structure fail, especially in depth ≥2 settings.
• Leveraging the fact that trace ideals are preserved under module isomorphism, the paper reduces the problem to studying principal ideals and their trace properties.
• Analyzing completions and their behavior under the Lindo–Pande condition, the paper constructs counterexamples to show that the condition does not always ascend under completion, especially in depth ≥2 rings.

Experimental results

Research questions

• RQ1 When is every ideal of a commutative noetherian local ring isomorphic to a trace ideal?
• RQ2 What characterizes local rings where all ideals are isomorphic to trace ideals, particularly in depth 0, 1, and higher dimensions?
• RQ3 How does the Lindo–Pande condition behave under completion, especially in rings of depth ≥2?
• RQ4 What is the relationship between factoriality and the property that all ideals are isomorphic to trace ideals?
• RQ5 Which singularities (e.g. hypersurfaces, A_n singularities) satisfy the Lindo–Pande condition?

Key findings

• A commutative noetherian local ring satisfies the Lindo–Pande condition (every ideal is isomorphic to a trace ideal) if and only if it is artinian Gorenstein, a 1-dimensional hypersurface of multiplicity at most 2, or a unique factorization domain.
• For local rings of depth 0, the Lindo–Pande condition is equivalent to being artinian Gorenstein, and all five conditions (trace ideal, isomorphic to trace ideal, etc.) are equivalent.
• In local rings of depth 1, the Lindo–Pande condition holds if and only if the ring is a 1-dimensional hypersurface of multiplicity ≤2, or its completion is an $(A_n)$-singularity for $0 \leq n \leq \infty$, with full equivalence under algebraically closed residue fields of characteristic 0.
• In rings where all maximal ideals have height ≥2, every ideal is isomorphic to a trace ideal if and only if the ring is a product of unique factorization domains.
• The Lindo–Pande condition does not necessarily ascend along completion: there exist factorial local rings of depth 2 whose completions are not factorial and thus fail the condition.
• The paper constructs a counterexample using Ogoma’s example of a 2-dimensional factorial local ring without a canonical module, showing that $R$ satisfies the Lindo–Pande condition while $\widehat{R}$ does not.