Left--right Transfer for $C4^{\ast}$-Rings Beyond the Regular Case
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917462423896064 |
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| author | Gokavarapu, Chandrasekhar |
| author_facet | Gokavarapu, Chandrasekhar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04053 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Left--right Transfer for $C4^{\ast}$-Rings Beyond the Regular Case Gokavarapu, Chandrasekhar Rings and Algebras 16D70, 16D80, 16E50, 16P20, 16U99 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. |
| title | Left--right Transfer for $C4^{\ast}$-Rings Beyond the Regular Case |
| topic | Rings and Algebras 16D70, 16D80, 16E50, 16P20, 16U99 |
| url | https://arxiv.org/abs/2605.04053 |