Separable functors and firm modules

Patrik Lundstr\"om

arxiv: 2602.13417 · v3 · pith:U2DYLBJNnew · submitted 2026-02-13 · 🧮 math.RA · math.RT

Separable functors and firm modules

Patrik Lundstr\"om This is my paper

Pith reviewed 2026-05-15 22:35 UTC · model grok-4.3

classification 🧮 math.RA math.RT

keywords separable functorsfirm modulesnonunital ringsMaschke theoremgroup ringssemisimplicityring extensions

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The pith

Firm modules over nonunital rings support separable functors and a locally unital Maschke theorem for group rings.

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

The paper develops a theory of separable ring extensions and separable functors for nonunital rings, carried out entirely in the category of firm modules. It establishes nonunital analogues of classical theorems on functorial separability and semisimplicity. These analogues are then used to obtain a version of Maschke's theorem that holds locally unital for group rings. A sympathetic reader cares because many rings arising in algebra and representation theory lack multiplicative units, and the firm-module setting removes the need for hidden unit assumptions when extending standard results.

Core claim

In the category of firm modules over a nonunital ring, the notions of separable ring extensions and separable functors can be defined directly, yielding nonunital analogues of the classical results on functorial separability and semisimplicity, which in turn produce a locally unital version of Maschke's theorem for group rings.

What carries the argument

Firm modules over a nonunital ring, the ambient category in which separable functors and separable extensions are defined without requiring the ring to possess a multiplicative identity.

If this is right

Separable extensions of nonunital rings exist and behave as expected inside the firm-module category.
Semisimplicity criteria for modules carry over directly to the nonunital setting via firm modules.
Group rings over nonunital coefficient rings admit a Maschke-type theorem that splits representations locally with respect to units.

Where Pith is reading between the lines

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

The same firm-module framework may support nonunital versions of other classical results such as the Artin-Wedderburn theorem.
Firm modules could serve as a uniform language for comparing unital and nonunital cases across different branches of ring theory.

Load-bearing premise

The category of firm modules over a nonunital ring is sufficiently rich and well-behaved to carry the classical definitions and proofs of separability and semisimplicity without extra restrictions imposed by the absence of a unit.

What would settle it

Existence of a nonunital ring and a functor on its firm modules that satisfies the classical definition of separability yet fails to obey the expected preservation properties for semisimplicity or Maschke-type splittings.

read the original abstract

We develop a theory of separable ring extensions and separable functors for nonunital rings in the setting of firm modules. We prove nonunital analogues of classical results on functorial separability and semisimplicity, and apply these results to obtain a locally unital version of Maschke's theorem for group rings.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript develops a theory of separable ring extensions and separable functors for nonunital rings in the setting of firm modules. It proves nonunital analogues of classical results on functorial separability and semisimplicity, and applies these results to obtain a locally unital version of Maschke's theorem for group rings.

Significance. If the results hold, the work supplies a coherent nonunital extension of separability and semisimplicity via the standard firm-module category, together with a concrete application to group rings. This framework is likely to be useful for further study of nonunital algebras and their representations.

major comments (1)

The central development assumes that the category of firm modules over a nonunital ring is sufficiently rich to carry the classical proofs of functorial separability and semisimplicity without hidden restrictions. An explicit verification or lemma confirming that the usual embedding of unital rings into the nonunital theory preserves all required exactness and splitting properties would make this load-bearing step fully transparent.

minor comments (2)

Notation for the firm-module functor and the associated separability idempotent should be introduced with a short table or diagram in the preliminary section to aid readability.
The statement of the locally unital Maschke theorem would benefit from an explicit comparison (in a remark) with the classical unital version, highlighting precisely which hypotheses are relaxed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

Referee: The central development assumes that the category of firm modules over a nonunital ring is sufficiently rich to carry the classical proofs of functorial separability and semisimplicity without hidden restrictions. An explicit verification or lemma confirming that the usual embedding of unital rings into the nonunital theory preserves all required exactness and splitting properties would make this load-bearing step fully transparent.

Authors: We agree that an explicit verification would improve the transparency of the central development. While the proofs of the nonunital analogues proceed directly in the firm-module setting and the embedding of unital rings is functorial by construction, we will add a short lemma in the revised version that confirms the embedding preserves the exactness and splitting properties needed for functorial separability and semisimplicity. This lemma will be placed early in the development of the main results. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper develops a theory of separable ring extensions and separable functors for nonunital rings via the firm modules category, proves direct nonunital analogues of classical functorial separability and semisimplicity results, and applies them to obtain a locally unital Maschke theorem for group rings. All steps rely on standard definitions of firm modules (the established setting for nonunital rings) together with self-contained proofs that extend classical unital results via the usual embedding; no predictions reduce to fitted parameters, no quantities are defined in terms of themselves, and no load-bearing self-citations or imported uniqueness theorems appear. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard categorical and module-theoretic axioms plus the new definition of firm modules; no free parameters or invented entities are visible from the abstract.

axioms (1)

standard math Standard axioms of rings, modules, and functors in abelian categories
Invoked implicitly when defining separability and semisimplicity analogues.

pith-pipeline@v0.9.0 · 5327 in / 1181 out tokens · 58192 ms · 2026-05-15T22:35:34.759045+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

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unclear
Relation between the paper passage and the cited Recognition theorem.

We prove nonunital analogues of classical results on functorial separability and semisimplicity, and apply these results to obtain a locally unital version of Maschke's theorem for group rings.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3. Suppose that f:B→A is a left firm ring homomorphism. Then A/B is separable if and only if the restriction functor Res_f : AFMod→BFMod is separable.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Separability for relative extensions of object unital strongly groupoid graded rings
math.RA 2026-05 unverdicted novelty 6.0

R/R_Δ is separable iff for each object e there exists f in its component and r in the centralizer of R_Λ such that the relative trace tr^f(r) equals the local identity.