A Comparative Study of Homological Dimensions in Commutative and Non Commutative Rings
DOI:
https://doi.org/10.31305/trjtm2023.v03.n03.005Keywords:
Homological dimensions, Commutative and non-commutative rings, Module theoryAbstract
Homological dimensions serve as essential invariants in algebra, providing insight into the structural complexity of rings and modules and offering a framework for comparing different algebraic systems. This paper presents a detailed comparative study of homological dimensions in commutative and non commutative rings, emphasizing the theoretical distinctions that arise due to the presence or absence of commutativity. Key homological invariants, including projective dimension, injective dimension, flat dimension, and global dimension, are analyzed to elucidate their role in module theory and ring classification. The study explores how non-commutativity introduces asymmetry between left and right module categories, resulting in divergent homological behaviors that do not appear in commutative contexts. It further examines the impact of these differences on the application of classical homological techniques, including resolutions and dimension theoretic arguments. By synthesizing established results from homological algebra, category theory, and ring theory, the paper provides a comprehensive framework for understanding the comparative structure of modules over commutative and non commutative rings. The findings reveal that homological dimensions are highly sensitive to the underlying algebraic structure, and non commutative rings often exhibit richer, more complex homological phenomena. These insights not only enhance our understanding of algebraic invariants but also have implications for related areas such as representation theory, module classification, and computational algebra. Overall, this work offers a rigorous theoretical foundation for further exploration of homological dimensions and serves as a guide for researchers studying the interplay between ring structure and module-theoretic properties.
References
Auslander, M., & Buchsbaum, D. A. (1957). Homological dimension in Noetherian rings. Proceedings of the National Academy of Sciences, 43(7), 548–555.
Bass, H. (1960). Injective dimension in Noetherian rings. Transactions of the American Mathematical Society, 102(2), 18–225.
Cartan, H., & Eilenberg, S. (1956). Homological Algebra. Princeton University Press.
Rotman, J. J. (2009). An Introduction to Homological Algebra (2nd ed.). Springer.
Holm, H. (2004). Gorenstein Homological Dimensions. Journal of Pure and Applied Algebra, 189(1–3), 167–193.
Chen, X. W., & Xi, C. (2011). Global dimension of non-commutative rings. Journal of Algebra, 329(1), 1–18.
Iyengar, S., & Krause, H. (2006). The derived category of a non-commutative ring. Journal of Algebra, 292(1), 1–25.
Enochs, E. E., & Jenda, O. M. G. (2000). Relative Homological Algebra. Walter de Gruyter.
Lam, T. Y. (2001). A First Course in Noncommutative Rings (2nd ed.). Springer.
Weibel, C. A. (1994). An Introduction to Homological Algebra. Cambridge University Press.
Matsumura, H. (1989). Commutative Ring Theory. Cambridge University Press.
McConnell, J. C., & Robson, J. C. (2001). Noncommutative Noetherian Rings (Revised Ed.). AMS Graduate Studies in Mathematics, 30.
Rotman, J. J. (2008). Advanced Modern Algebra (2nd ed.). American Mathematical Society.
Lam, T. Y. (1999). Lectures on Modules and Rings. Springer.
Gelfand, S. I., & Manin, Y. I. (1996). Methods of Homological Algebra. Springer.
Bruns, W., & Herzog, J. (1993). Cohen-Macaulay Rings. Cambridge University Press.
Auslander, M. (1967). Modules over unramified regular local rings. Illinois Journal of Mathematics, 11(3), 447–452.
Benson, D. J. (1991). Representations and Cohomology I: Basic Representation Theory of Finite Groups and Associative Algebras. Cambridge University Press.
Happel, D. (1988). Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. Cambridge University Press.
Downloads
Published
Issue
Section
Deprecated: json_decode(): Passing null to parameter #1 ($json) of type string is deprecated in /home/u495429466/domains/technoreview.co.in/public_html/plugins/generic/citations/CitationsPlugin.php on line 68
