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arxiv: 2604.26712 · v1 · submitted 2026-04-29 · 🧮 math.AC

Recognition: unknown

k[x]-modules and Core-Nilpotent endomorphisms

Diego Alba Alonso, Javier S\'anchez Gonz\'alez

Pith reviewed 2026-05-07 11:06 UTC · model grok-4.3

classification 🧮 math.AC
keywords core-nilpotent endomorphismsk[x]-modulesDrazin inversegeneralized inversesendomorphismsvector spacesnilpotent operatorsdecompositions
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The pith

Endomorphisms on arbitrary vector spaces admit core-nilpotent decompositions via their induced k[x]-module structures.

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

The paper establishes that core-nilpotent endomorphisms form the largest class of endomorphisms on any vector space that can be decomposed into a sum of two endomorphisms mirroring the core-nilpotent decomposition of matrices. By equipping the vector space with a k[x]-module structure defined by the endomorphism, the authors characterize these endomorphisms and derive the decomposition. This framework also introduces a generalized inverse for such endomorphisms that coincides with the Drazin inverse under appropriate conditions. The results extend classical linear algebra results from finite-dimensional matrices to endomorphisms on spaces of arbitrary dimension.

Core claim

Core-nilpotent endomorphisms over an arbitrary vector space form the largest subset of the ring of endomorphisms over that arbitrary vector space which admit a decomposition as sum of two endomorphisms satisfying the analogous properties as the well known core-nilpotent decomposition of matrices. The paper presents a new description of core-nilpotent endomorphisms using the k[x]-module structure they define in the base vector space. This approach provides a new generalized inverse that restricts to the well known Drazin inverse under certain conditions, and presents a generalized core-nilpotent decomposition for endomorphisms over arbitrary vector spaces.

What carries the argument

The k[x]-module structure induced by an endomorphism on the underlying vector space, which encodes the action of the endomorphism as multiplication by x and permits module-theoretic statements of the core and nilpotent conditions.

Load-bearing premise

That the algebraic properties of core and nilpotent summands in the finite-dimensional matrix decomposition transfer directly to endomorphisms on possibly infinite-dimensional spaces when rephrased in terms of the induced k[x]-module structure.

What would settle it

An explicit endomorphism T on an infinite-dimensional vector space such that the module-theoretic core part plus nilpotent part fails to recover T or violates one of the defining relations (core part invertible on its image, nilpotent part nilpotent, and mutual annihilation) while still satisfying the classical matrix-style conditions, or conversely an endomorphism inside the claimed largest class that admits no such decomposition.

read the original abstract

Core-nilpotent endomorphisms over an arbitrary vector space form the largest subset of the ring of endomorphisms over that arbitrary vector space which admit a decomposition as sum of two endomorphisms satisfying the analogous properties as the well known core-nilpotent decomposition of matrices. In this paper we present a new description of core-nilpotent endomorphisms using the $k[x]-$module structure they define in the base vector space. Moreover, our approach provides us with a ``new'' generalized inverse that restricts to the well known Drazin inverse under certain conditions. Similarly, we present a generalized core-nilpotent decomposition for endomorphisms over arbitrary vector spaces.

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

0 major / 3 minor

Summary. The paper claims that core-nilpotent endomorphisms on an arbitrary vector space V, defined via the induced k[x]-module structure, form the largest subset of End(V) admitting a decomposition T = C + N where C is core (invertible on its image) and N is nilpotent, with properties analogous to the finite-dimensional matrix case. It further asserts that this yields a new generalized inverse restricting to the Drazin inverse when the ascent is finite, and provides a generalized core-nilpotent decomposition for arbitrary endomorphisms.

Significance. If the central claims hold, the module-theoretic characterization offers a clean way to handle core-nilpotent operators uniformly in finite and infinite dimensions, which could aid work on generalized inverses and spectral theory over arbitrary fields. The explicit link to the Drazin inverse and the maximality statement are potentially useful contributions if the proofs are complete and the 'new' inverse is shown to be distinct in a meaningful way.

minor comments (3)
  1. The abstract and introduction should explicitly recall the standard definition of the core-nilpotent decomposition (including the commuting condition and the precise meaning of 'core') before stating the generalization, to make the analogy clear to readers.
  2. Clarify the precise construction of the new generalized inverse in the relevant theorem; it is not immediately obvious from the abstract whether it coincides with an existing construction (e.g., via the module decomposition) or offers a genuinely different formula.
  3. In the section presenting the k[x]-module structure, include a brief remark on how the finite-ascent condition translates into the module language (e.g., the module being a direct sum of a free summand and a torsion summand of finite length).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. We appreciate the acknowledgment of the potential utility of the module-theoretic approach to core-nilpotent endomorphisms in both finite and infinite dimensions, as well as the link to the Drazin inverse. Since the report contains no specific major comments or criticisms, we have no point-by-point rebuttals to provide. We will incorporate any minor editorial improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central contribution is a module-theoretic characterization of core-nilpotent endomorphisms on arbitrary vector spaces via the k[x]-module structure induced by the endomorphism. This characterization is used to establish the maximality property for the core-nilpotent decomposition and the associated generalized inverse (restricting to the Drazin inverse when ascent is finite). The derivation relies on standard definitions of k[x]-modules, direct sums, and the finite-ascent condition (ker T^m = ker T^{m+1} and im T^m = im T^{m+1}), which are independently known to be necessary and sufficient for the decomposition V = im(T^m) ⊕ ker(T^m). No step reduces by construction to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation; the results are self-contained against external benchmarks in linear algebra and module theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The paper introduces these concepts based on the module structure, relying on prior knowledge of Drazin inverse and core-nilpotent decompositions for matrices.

axioms (2)
  • standard math Vector spaces are defined over a field k
    Fundamental to the setup of endomorphisms.
  • domain assumption An endomorphism T defines a k[x]-module structure on V by x*v = T(v)
    This is the key construction mentioned in the abstract.
invented entities (2)
  • Generalized core-nilpotent decomposition no independent evidence
    purpose: To extend the matrix decomposition to arbitrary vector spaces
    Presented as a new generalization.
  • New generalized inverse no independent evidence
    purpose: To provide an inverse that restricts to Drazin inverse
    Claimed as new but with restriction condition.

pith-pipeline@v0.9.0 · 5397 in / 1401 out tokens · 148468 ms · 2026-05-07T11:06:38.479650+00:00 · methodology

discussion (0)

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Reference graph

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