Author
A Sivasankari, K Thavamani
Keywords
Ring Theory; Ideals; Euclidean Rings; Quotient Rings; Polynomial Rings; Abstract Algebra
Abstract
Ring theory is a central area of abstract algebra that generalizes arithmetic operations and polynomial algebra through algebraic structures equipped with two binary operations: addition and multiplication. This paper presents a systematic study of ring theory, beginning with basic definitions and progressing through important classes of rings such as integral domains, fields, Euclidean rings, and polynomial rings. Special emphasis is given to ideals, quotient rings, maximal ideals, and homeomorphisms, which play a vital role in understanding the structural properties of rings. The paper also discusses Euclidean rings and unique factorization, illustrating how classical number-theoretic results emerge naturally from ring-theoretic concepts. The study aims to provide a clear and rigorous foundation for students and researchers interested in modern algebra and its applications in mathematics, computer science, and related disciplines.
References
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[2] Atiyah, M. F., & Macdonald, I. G. (1969). Introduction to Commutative Algebra. AddisonWesley Publishing Company.
[3] Eisenbud, D. (1995). Commutative Algebra: With a View Toward Algebraic Geometry. SpringerVerlag.
[4] Lang, S. (2002). Algebra (3rd ed.). SpringerVerlag.
[5] Matsumura, H. (1989). Commutative Ring Theory. Cambridge University Press.
[6] Rotman, J. J. (2009). Advanced Modern Algebra. American Mathematical Society.
[7] Galaz-García, F., Kerin, M., Radeschi, M., Wiemeler, M.: Torus orbifolds, slice-maximal torus actions, and rational ellipticity. Int. Math. Res. Not. IMRN 18, 5786–5822 (2018)
[8] Guillemin, V., Sabatini, S., Zara, C.: Equivariant -theory of GKM bundles. Ann. Global Anal. Geom. 43(1), 31–45 (2013)
[9] Aberbach, I.M., Enescu, F.: The structure of F-pure rings. Math. Z. 250(4), 791–806 (2005)
[10] Akesseh, S.: Ideal containments under flat extensions. J. Algebra 492, 44–51 (2017).
Received : 11 December 2025
Accepted : 17 March 2026
Published : 27 March 2026
DOI: 10.30726/esij/v13.i1.2026.131003