Bibliographic Information: Bennett-Tennenhaus, R. (2024). Linear relations over commutative rings [Preprint]. arXiv:2410.05888v1.
Research Objective: This paper aims to analyze the category of linear relations over a commutative ring through the lens of Kronecker representations. The author investigates various properties of this category, including its closure properties, the existence of envelopes and covers, and its relationship to torsion theories.
Methodology: The author employs techniques from abstract algebra, particularly module theory and representation theory. Key concepts utilized include torsion pairs, injective and projective modules, Morita contexts, and the functorial filtrations classification method.
Key Findings: The paper establishes the equivalence of the category of linear relations over a commutative ring to a specific subcategory of Kronecker representations. This subcategory is characterized by the injectivity of a particular morphism. The author further demonstrates that this subcategory is closed under various operations, including subobjects, extensions, limits, and coproducts. Additionally, the paper explores the existence of envelopes and covers within this category and relates it to torsion theories.
Main Conclusions: By establishing the connection between linear relations and Kronecker representations, the paper provides a new perspective for studying linear relations. The closure properties and the existence of envelopes and covers in the identified subcategory offer valuable tools for further investigations. The paper's findings contribute to a deeper understanding of the structure and properties of linear relations over commutative rings.
Significance: This research holds significance within the field of representation theory by providing a novel framework for analyzing linear relations. The established equivalence and the explored properties offer a new avenue for studying these mathematical objects and their applications in various areas of mathematics.
Limitations and Future Research: The paper primarily focuses on linear relations over commutative rings. Future research could explore the generalization of these results to non-commutative rings or investigate the applications of the established framework in specific areas where linear relations play a crucial role.
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by Raphael Benn... at arxiv.org 10-10-2024
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