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arxiv: 2605.04053 · v1 · submitted 2026-03-09 · 🧮 math.RA

Left--right Transfer for C4^(ast)-Rings Beyond the Regular Case

Chandrasekhar Gokavarapu (Department of Mathematics , Government College (Autonomous) , Rajahmundry , Andhra Pradesh , India) This is my paper

Pith reviewed 2026-05-15 12:56 UTC · model grok-4.3

classification 🧮 math.RA
keywords C4*-ringsleft-right symmetrytransfer mechanismsorthogonal decompositioncorner controlsquare-free phenomenaring theory
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The pith

Right C4* conditions transfer to the left under weaker-than-regularity hypotheses using orthogonal decomposition and corner control.

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

The paper shows that regularity guarantees left-right symmetry for strongly C4* rings but is stronger than needed. It isolates a transfer mechanism, based on orthogonal decomposition, corner control, and summand-square-free separation, that forces right C4*-type conditions to imply their left counterparts. This matters because it separates cases where one-sided behavior automatically becomes two-sided from cases blocked by persistent one-sided square-free phenomena. The same mechanism is shown to survive matrix rings and full corner passages, placing the symmetry question in a stable framework without claiming full Morita invariance.

Core claim

Right C4*-rings, strongly right C4*-rings, and right semi-weak-CS C4*-rings satisfy left-right symmetry under hypotheses strictly weaker than regularity. The transfer proceeds via orthogonal decomposition, corner control, and summand-square-free separation rather than regular decomposition. Sufficient conditions for transfer are obtained, necessary obstruction patterns are identified (persistent one-sided square-free phenomena and breakdown of compatible corner decompositions), and the resulting principles remain stable under matrix and full-corner passage.

What carries the argument

The transfer mechanism driven by orthogonal decomposition, corner control, and summand-square-free separation, which carries right C4* conditions to the left side without invoking regularity.

If this is right

  • Right C4* conditions imply left C4* conditions whenever the weaker hypotheses hold.
  • One-sided C4* behavior forces two-sided behavior in the presence of compatible corner decompositions.
  • Transfer principles persist under passage to matrix rings and full corners when the hypotheses are preserved.
  • Failure of symmetry is structural and occurs precisely when one-sided square-free phenomena persist.
  • The symmetry problem splits into transfer, obstruction, and permanence parts that refine the regular case.

Where Pith is reading between the lines

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

  • The separation into transfer and obstruction cases offers a classification scheme that applies to wider families of rings than the regular ones previously studied.
  • The permanence results suggest that C4* symmetry questions can be studied in a Morita-stable setting even if full invariance is not yet established.
  • Explicit constructions of rings with controlled orthogonal decompositions could test which obstruction patterns are minimal.

Load-bearing premise

The hypotheses of orthogonal decomposition, corner control, and summand-square-free separation remain compatible and stable even when the ring is not regular.

What would settle it

A concrete ring satisfying the weaker hypotheses in which a right C4* condition holds but the corresponding left condition fails, with the failure traceable to an explicit one-sided square-free summand, would disprove the transfer claims.

read the original abstract

The first structural fact is that regularity is sufficient for left--right symmetry of the strongly \(C4^{\ast}\) condition. It is not necessary for the definition itself and is too strong for classification. The problem is therefore to determine which weaker hypotheses still force right \(C4^{\ast}\)-type conditions to pass to the left side, and which obstructions prevent such transfer. We study right \(C4^{\ast}\)-rings, strongly right \(C4^{\ast}\)-rings, and right semi-weak-CS \(C4^{\ast}\)-rings under hypotheses strictly weaker than regularity. The method does not repeat the regular decomposition argument. Instead, it isolates a transfer mechanism based on orthogonal decomposition, corner control, and summand-square-free separation. This yields sufficient conditions for left--right transfer beyond the regular case and also identifies necessary obstruction patterns. In particular, we determine settings in which one-sided \(C4^{\ast}\)-behavior forces two-sided \(C4^{\ast}\)-behavior, and settings in which the implication fails. A second aim is exact separation. We show that failure of transfer is structural, not accidental: it is encoded by persistent one-sided square-free phenomena and by the breakdown of compatible corner decompositions. This produces counterexample criteria rather than isolated examples. A third aim is permanence. The transfer principles are formulated so as to persist under matrix and full-corner passage whenever the relevant hypotheses are stable. Thus the symmetry problem is placed in a Morita-type framework, although full Morita invariance is not asserted for every \(C4^{\ast}\)-condition. The paper therefore divides the symmetry problem into transfer, obstruction, and permanence, and thereby sharpens the regular theory, delimits its range, and gives a classification scheme for the left--right behavior of \(C4^{\ast}\)-type 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

3 major / 1 minor

Summary. The manuscript claims that regularity is sufficient but not necessary for left-right symmetry of the strongly C4* condition. It develops a transfer mechanism for right C4*-rings, strongly right C4*-rings, and right semi-weak-CS C4*-rings under hypotheses strictly weaker than regularity, using orthogonal decomposition, corner control, and summand-square-free separation rather than repeating regular decomposition arguments. This yields sufficient conditions for left-right transfer, identifies necessary obstructions via persistent one-sided square-free phenomena and breakdown of compatible corner decompositions, and establishes permanence of the transfer principles under matrix and full-corner passage when the hypotheses are stable, placing the symmetry problem in a Morita-type framework and providing a classification scheme for left-right behavior of C4*-type rings.

Significance. If the results hold, the work would sharpen the regular theory of C4*-rings by delimiting its range, supplying structural counterexample criteria rather than isolated examples, and extending symmetry results to weaker hypotheses. The division into transfer, obstruction, and permanence sections offers a coherent framework that could aid classification in ring theory and support further Morita-type investigations, though full Morita invariance is not claimed.

major comments (3)
  1. [Transfer mechanism] The central claim that the transfer mechanism isolates left-right passage via orthogonal decomposition, corner control, and summand-square-free separation without repeating regular decomposition arguments (as stated in the abstract) requires explicit contrast in the transfer section: it must be shown that these tools remain independent of regularity and do not implicitly rely on regular summand properties to establish the sufficient conditions for transfer.
  2. [Obstruction patterns] The assertion that failure of transfer is structural and encoded by persistent one-sided square-free phenomena (abstract, obstruction patterns) needs concrete criteria or constructions demonstrating the breakdown of compatible corner decompositions; without such, the distinction from accidental failure remains unverified and load-bearing for the classification scheme.
  3. [Permanence] The permanence claim under matrix and full-corner passage (abstract) depends on stability of the weaker hypotheses; the manuscript must specify exact stability conditions for orthogonal decomposition and summand-square-free separation to rigorously support the Morita-type framework.
minor comments (1)
  1. The terms 'strongly C4*', 'semi-weak-CS C4*', and 'summand-square-free' should be defined with precise ring-theoretic conditions in the introduction or preliminary section to aid readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [Transfer mechanism] The central claim that the transfer mechanism isolates left-right passage via orthogonal decomposition, corner control, and summand-square-free separation without repeating regular decomposition arguments (as stated in the abstract) requires explicit contrast in the transfer section: it must be shown that these tools remain independent of regularity and do not implicitly rely on regular summand properties to establish the sufficient conditions for transfer.

    Authors: The transfer section is formulated exclusively under the weaker hypotheses, avoiding any regular decomposition steps. To provide the explicit contrast requested, we will add a dedicated paragraph verifying that orthogonal decomposition, corner control, and summand-square-free separation operate independently of regularity, with a direct comparison to the regular case showing where the weaker tools suffice. revision: yes

  2. Referee: [Obstruction patterns] The assertion that failure of transfer is structural and encoded by persistent one-sided square-free phenomena (abstract, obstruction patterns) needs concrete criteria or constructions demonstrating the breakdown of compatible corner decompositions; without such, the distinction from accidental failure remains unverified and load-bearing for the classification scheme.

    Authors: The obstruction section already identifies persistent one-sided square-free phenomena as the source of incompatible corner decompositions and supplies criteria for when transfer fails. To address the concern directly, we will expand this with additional explicit constructions illustrating the breakdown, thereby confirming the structural character of the failure and strengthening the classification scheme. revision: yes

  3. Referee: [Permanence] The permanence claim under matrix and full-corner passage (abstract) depends on stability of the weaker hypotheses; the manuscript must specify exact stability conditions for orthogonal decomposition and summand-square-free separation to rigorously support the Morita-type framework.

    Authors: We agree that explicit stability conditions are needed for rigor. In the revised permanence section we will state the precise conditions under which orthogonal decomposition and summand-square-free separation remain stable under matrix and full-corner passage, thereby supporting the Morita-type framework as claimed. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper isolates a transfer mechanism via orthogonal decomposition, corner control, and summand-square-free separation under hypotheses strictly weaker than regularity. These are presented as independent structural tools that do not reduce to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The division into transfer, obstruction, and permanence is internally consistent on the stated terms without any quoted equation or premise collapsing to its own inputs by construction. No uniqueness theorems or ansatzes are smuggled via prior self-work in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard ring and module axioms from prior literature; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract description.

axioms (1)
  • standard math Standard axioms of rings and modules, including properties of orthogonal decompositions and corner rings
    Invoked as background for the transfer mechanism and obstruction analysis.

pith-pipeline@v0.9.0 · 5662 in / 1146 out tokens · 43290 ms · 2026-05-15T12:56:08.706270+00:00 · methodology

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Reference graph

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