arxiv: 2605.12459 · v1 · submitted 2026-05-12 · 🧮 math.RA

Recognition: 2 theorem links

· Lean Theorem

Trace ideals and uniserial modules

Dolors Herbera, Pavel P\v{r}\'i hoda

Pith reviewed 2026-05-13 02:20 UTC · model grok-4.3

classification 🧮 math.RA

keywords trace idealsuniserial modulesendomorphism ringsprojective modulesserial modulesone-sided idealsmodule lifting

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The pith

Endomorphism rings of uniserial modules admit trace ideals of projective right modules that fail to be trace ideals of the corresponding left modules.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates trace ideals attached to projective modules over the endomorphism ring of a uniserial module. It supplies an intrinsic description of those trace ideals that arise on the right but not on the left, together with their basic properties. The same lifting theory for projective modules modulo a trace ideal is then applied to recover, by a different route, a direct summand of a serial module that is not itself serial.

Core claim

Over the endomorphism ring of a uniserial module, there exist trace ideals of projective right modules that are not trace ideals of projective left modules. These ideals admit an intrinsic description in terms of the underlying uniserial module, and the associated lifting technique for projective modules modulo the trace ideal yields an alternative construction of a non-serial direct summand inside a serial module.

What carries the argument

The trace ideal of a projective module over the endomorphism ring of a uniserial module, which distinguishes right and left behavior and supports the lifting of projectives modulo the ideal.

If this is right

These rings furnish concrete examples of projective modules whose trace ideals are strictly one-sided.
The intrinsic description classifies the trace ideals according to properties of the uniserial module.
Lifting projective modules modulo such a trace ideal produces a direct summand of a serial module that is not serial.
The same lifting machinery applies uniformly to both the asymmetry result and the construction of non-serial summands.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same description may extend to other classes of modules whose endomorphism rings are close to uniserial.
The lifting approach could shorten existing proofs that rely on direct constructions of non-serial summands.
Questions about when trace ideals become two-sided reduce to conditions on the underlying uniserial module.

Load-bearing premise

The endomorphism ring of a uniserial module is assumed to behave in a way that permits trace ideals to be one-sided and allows projective modules to lift modulo those ideals.

What would settle it

An explicit uniserial module whose endomorphism ring has a trace ideal of a projective right module that is nevertheless also the trace ideal of a projective left module would refute the claimed asymmetry.

read the original abstract

We thoroughly investigate the trace ideals of projective modules over the endomorphism ring of a uniserial module. After the work of Dubrovin and Puninski, it is known that this class of rings provides examples of trace ideals of projective right modules that are not trace ideals of projective left modules. In this paper we further investigate when this happens, giving an intrinsic description of such trace ideals and their properties. We also use the theory associated to lifting projective modules modulo a trace ideal to give an alternative approach to Puninski's construction of a direct summand of a serial module that is not serial.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a clean intrinsic description of asymmetric trace ideals over endomorphism rings of uniserial modules and a lifting-based alternative to Puninski's non-serial summand construction.

read the letter

This paper gives a clean intrinsic description of asymmetric trace ideals over endomorphism rings of uniserial modules and a lifting-based alternative to Puninski's non-serial summand construction. It stays tightly focused on the setting developed by Dubrovin and Puninski, where these rings already supply examples of trace ideals that behave differently on the right and left for projective modules. The new work spells out when that asymmetry occurs and lists the properties that follow from it, without needing to fix a particular module or ring at the start. The second part recasts Puninski's direct summand example by lifting projective modules modulo the trace ideal, which looks like a shorter route once the lifting machinery is in place. Both pieces read as genuine extensions rather than restatements. The abstract is direct about the dependence on earlier results, and the stated goals line up with what the prior literature already supports. No internal contradictions appear in the outline. The main limitation is scope. The results organize a narrow corner of module theory and do not claim wider consequences or resolve standing questions outside this class of rings. If the intrinsic description ends up being as technical as the original constructions in practice, it may see limited uptake. Still, the claims are modest and the approach is consistent with the cited foundations. This is for readers already working on uniserial modules, endomorphism rings, or trace ideals in noncommutative settings. Someone who knows Dubrovin-Puninski will find the new descriptions and the alternative construction useful to check. It is worth sending to a serious referee; the contributions are concrete enough to be verified against the literature and the proofs can be examined in detail.

Referee Report

0 major / 2 minor

Summary. The paper investigates trace ideals of projective modules over the endomorphism ring of a uniserial module. Building on Dubrovin and Puninski, it provides an intrinsic description of those trace ideals that arise for projective right modules but not for projective left modules, along with their properties. It further applies the theory of lifting projective modules modulo a trace ideal to give an alternative construction of a direct summand of a serial module that is not itself serial.

Significance. If the results hold, the work advances the study of trace ideals in noncommutative ring theory by supplying intrinsic characterizations that clarify the right-left asymmetry for projective modules over endomorphism rings of uniserial modules. The alternative lifting-based approach to Puninski's non-serial summand construction offers a potentially more conceptual route that may generalize or simplify related arguments in serial module theory.

minor comments (2)

The abstract refers to 'thoroughly investigate' and 'further investigate when this happens' without naming the principal theorems or the precise conditions under which the right-but-not-left phenomenon occurs; adding one or two explicit statements of the main results would improve readability.
Notation for trace ideals (e.g., distinctions between right and left versions) should be introduced with a short table or explicit list of symbols in the preliminaries to avoid ambiguity when the same ideal is viewed from both sides.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work on trace ideals of projective modules over endomorphism rings of uniserial modules and for recommending minor revision. No specific major comments or criticisms were provided in the report, so we have no points requiring detailed rebuttal or revision at this stage. We appreciate the recognition of the intrinsic descriptions and the alternative approach to Puninski's construction.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper cites external prior work by Dubrovin and Puninski as the source of known examples of asymmetric trace ideals over endomorphism rings of uniserial modules and the associated lifting theory. It then derives an intrinsic description of right-but-not-left trace ideals and an alternative construction of a non-serial direct summand, both of which are presented as new contributions resting on those external foundations rather than re-deriving or re-fitting the inputs. No equations or claims reduce by construction to the paper's own definitions, no self-citations are load-bearing, and the derivation chain remains self-contained against the cited benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard axioms of associative rings and modules together with the prior results of Dubrovin and Puninski; no free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Standard axioms of ring theory and module theory over associative rings
The investigation operates entirely within the established framework of modules and their endomorphism rings.

pith-pipeline@v0.9.0 · 5387 in / 1220 out tokens · 47893 ms · 2026-05-13T02:20:51.103232+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
Theorem A (Theorems 5.5 and 6.8): T = SfS ⊆ K is the trace of a countably generated projective pure right ideal P; L = SgS ⊆ I is the trace of a countably generated projective pure left ideal Q; any projective right S-module is free ⊕ copies of P.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear
Proposition 3.5 and Corollary 2.3: traces of projective modules are determined modulo J(S); only 0, S, T (resp. L) appear when S/J(S) ≅ D1 × D2.

Reference graph

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