arxiv: 2604.13911 · v2 · submitted 2026-04-15 · 🧮 math.AC · math.RA

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A constructive proof of Orzech's theorem

Darij Grinberg

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Pith reviewed 2026-05-10 12:08 UTC · model grok-4.3

classification 🧮 math.AC math.RA

keywords Orzech's theoremconstructive proofCayley-Hamilton theoremfinitely generated modulecommutative ringsurjective homomorphismmodule isomorphism

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The pith

Any surjective homomorphism from a submodule of a finitely generated module over a commutative ring onto the module is an isomorphism.

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

The paper establishes a constructive proof of Orzech's theorem from 1971. The theorem states that if A is a commutative ring with unity and M is a finitely generated A-module, then any surjective A-module homomorphism from a submodule of M to M is necessarily an isomorphism. A reader would care because the result gives a precise control on how submodules can map onto the whole module, and the constructive nature means the proof yields an explicit way to recover the inverse map. The argument applies the Cayley-Hamilton theorem to an endomorphism built from the given surjection to produce the required identity relation.

Core claim

Let A be a commutative ring with unity and M a finitely generated A-module. If N is any submodule of M and φ: N → M is a surjective A-linear map, then φ is an isomorphism. The proof is constructive and proceeds by using the Cayley-Hamilton theorem on a suitable endomorphism of M derived from φ and the inclusion of N.

What carries the argument

The Cayley-Hamilton theorem applied to an endomorphism of the finitely generated module M constructed from the surjective homomorphism.

If this is right

The identity map on M cannot factor through a proper submodule via a surjective map.
Any surjective endomorphism of a finitely generated module over such a ring is automatically an isomorphism.
The result supplies an explicit algebraic identity that forces the kernel of φ to vanish.
The proof works uniformly for all finitely generated modules without additional finiteness or projectivity hypotheses.

Where Pith is reading between the lines

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

The same Cayley-Hamilton device may adapt to prove related statements about exact sequences or Fitting ideals in the same constructive style.
Because the proof is explicit, it yields a finite algorithm to decide injectivity once a presentation of M is given.
The argument highlights how determinant-based identities can replace choice principles in module theory.

Load-bearing premise

The ring must be commutative and possess a multiplicative identity so that the Cayley-Hamilton theorem applies to endomorphisms of finitely generated modules.

What would settle it

An explicit example of a commutative ring A with unity, a finitely generated A-module M, a submodule N, and a surjective homomorphism φ: N → M whose kernel is nonzero.

read the original abstract

Let $A$ be a commutative ring with unity, and $M$ a finitely generated $A$-module. In 1971, Morris Orzech showed that any surjective $A$-module homomorphism from a submodule of $M$ to $M$ must be an isomorphism. We give a constructive proof of this fact using the Cayley--Hamilton theorem.

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

Grinberg supplies a short, explicit constructive proof of Orzech's theorem by lifting the map to a free module on generators and applying Cayley-Hamilton there.

read the letter

Grinberg gives a constructive proof of Orzech's theorem: over a commutative ring with unit, any surjective homomorphism from a submodule of a finitely generated module M onto M is an isomorphism. The argument lifts the endomorphism to an endomorphism of a free module A^n on a chosen generating set, applies the standard Cayley-Hamilton theorem to obtain a monic polynomial relation, and verifies that the relation descends to M and forces the original map to be injective (hence an isomorphism since it is already surjective). This is the main new element: an explicit, choice-free presentation that was not in Orzech's 1971 paper. The write-up is short and stays within elementary commutative algebra, which makes the steps easy to follow and potentially useful for explicit computations or formalization work. The lifting step is canonical once generators are fixed, and the descent works because the submodule relations are preserved under the polynomial action. No circularity appears; Cayley-Hamilton is invoked as an external, standard result for matrices. A minor soft spot is that the proof is tied to a specific choice of generators, but this does not affect correctness or constructivity of the existence claim. The citation pattern is minimal and appropriate, referencing only the original theorem and basic facts about Cayley-Hamilton. This note is for readers who want an explicit version of the result, such as those working in constructive algebra or teaching module theory. It does not resolve open problems but makes a known fact more transparent. I would send it to peer review; the argument is self-contained and short enough for a referee to check in detail without much effort.

Referee Report

2 major / 2 minor

Summary. The paper presents a constructive proof of Orzech's 1971 theorem: if A is a commutative ring with unity and M is a finitely generated A-module, then any surjective A-linear map f: N → M with N ⊆ M is necessarily an isomorphism. The argument proceeds by applying the Cayley-Hamilton theorem to a suitable endomorphism induced by f after choosing a finite generating set for M.

Significance. The result is a classical fact in commutative algebra. A constructive proof that explicitly invokes only the standard Cayley-Hamilton theorem (rather than Nakayama's lemma or determinant tricks) could be useful for formalization in proof assistants and for algorithmic applications. The manuscript supplies the missing constructive details that the abstract alone leaves implicit.

major comments (2)

[§3] §3, lines 45-62: the lifting step from an arbitrary surjection f: N → M to an endomorphism of A^n is described only by reference to 'the usual presentation matrix'; an explicit matrix construction (or at least a reference to the precise lemma used) is needed to confirm that the monic polynomial obtained via Cayley-Hamilton descends to an annihilator relation on M itself.
[§4] §4, Eq. (7): the claim that the Cayley-Hamilton relation implies injectivity of f relies on the fact that the kernel of the surjection is annihilated by the constant term of the monic polynomial; this step is correct only after verifying that the relation is independent of the choice of generators, which is asserted but not proved in detail.

minor comments (2)

The notation for the submodule N is introduced without a dedicated symbol; using N consistently would improve readability.
Reference to the original Orzech paper is given only in the introduction; a precise citation (e.g., Orzech, J. Algebra 1971) should appear in the bibliography.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the helpful suggestions for improving clarity. We address each major comment below.

read point-by-point responses

Referee: §3, lines 45-62: the lifting step from an arbitrary surjection f: N → M to an endomorphism of A^n is described only by reference to 'the usual presentation matrix'; an explicit matrix construction (or at least a reference to the precise lemma used) is needed to confirm that the monic polynomial obtained via Cayley-Hamilton descends to an annihilator relation on M itself.

Authors: We agree that an explicit description of the lifting would strengthen the exposition. In the revised manuscript we will insert a short paragraph immediately after the choice of generators m_1,…,m_n for M, spelling out the construction: let n_i ∈ N be any lifts of the m_i under f; the A-linear map A^n → M sending the standard basis vector e_i to m_i factors through the surjection f: N → M, yielding an endomorphism φ of A^n whose matrix (with respect to the chosen basis) is the usual presentation matrix of M relative to these generators. Applying Cayley–Hamilton to φ then produces a monic polynomial whose constant term annihilates each m_i and hence the whole module M. This makes the descent of the relation fully explicit without relying on external lemmas. revision: yes
Referee: §4, Eq. (7): the claim that the Cayley-Hamilton relation implies injectivity of f relies on the fact that the kernel of the surjection is annihilated by the constant term of the monic polynomial; this step is correct only after verifying that the relation is independent of the choice of generators, which is asserted but not proved in detail.

Authors: The referee is correct that a brief verification of independence is desirable for full rigor. We will add one paragraph in §4 showing that if two different finite generating sets are chosen, the resulting monic polynomials p(t) and q(t) both annihilate M, so their constant terms both annihilate every element of M; consequently the kernel of f is annihilated by either constant term. Because the argument never uses more than the existence of some finite generating set, the injectivity conclusion is unaffected by the particular choice. This addition removes the asserted-but-unproved step while preserving the constructive character of the proof. revision: yes

Circularity Check

0 steps flagged

No circularity: external Cayley-Hamilton theorem applied to a standard module-theoretic argument

full rationale

The derivation invokes the Cayley-Hamilton theorem as an independent, externally established result (standard for endomorphisms of free finite-rank modules over commutative rings) and uses it to establish the injectivity of the given surjection. No equations or definitions in the paper reduce the target statement to a tautology, a fitted parameter, or a self-citation chain; the lifting from a general finitely generated module to a free module and the subsequent descent are part of the standard constructive application of the theorem rather than a redefinition of its inputs. The proof is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof rests on the standard Cayley-Hamilton theorem and basic definitions of modules and homomorphisms over commutative rings.

axioms (1)

standard math Cayley-Hamilton theorem holds for endomorphisms of finitely generated modules over commutative rings with unity
Invoked directly to construct the inverse or show injectivity in the proof.

pith-pipeline@v0.9.0 · 5335 in / 1030 out tokens · 43771 ms · 2026-05-10T12:08:08.179127+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 6 canonical work pages · 1 internal anchor

[1]

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work page doi:10.1080/00029890.1971.11992759 1971
[2]

Morris Orzech, Luis Ribes,Residual finiteness and the Hopf property in rings, Journal of Algebra15(1970), Issue 1, pp. 81–88. https://doi.org/10.1016/0021-8693(70)90087-6

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