A Topological Space Defined on Finite Rings Modulo Composite Number
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Abstract
For a composite number n, we explore a topological space on the ring of integers modulo n , we use basis sets formed by solutions to quadratic residue equations, where elements are units modulo n . This definition allows us to investigate the algebraic relationships among solutions, focusing on properties like clopen sets, closure, and interior operations. Additionally, we examine the continuity of mappings between the group of units in Z/nZ and quadratic residues, alongside the quotient topology induced on this group of units. A comparison of the separation properties of these topological spaces is also presented.
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References
Zariski, Oscar. The Topology of the Spectrum of an Algebraic Field. Transactions of the American Mathematical Society, vol. 44, no. 2, 1938, pp. 138–161.
Han, Sang-Eon, Saeid Jafari, and Jeong Min Kang. Topologies on Z_n that Are Not Homeomorphic to the n-Dimensional Khalimsky Topological Space. Mathematics 7.11 (2019): 1072.
Ireland, Kenneth, and Michael Rosen. A Classical Introduction to Modern Number Theory. 2nd ed., Springer-Verlag, 1990.
Cohen, Henri, S. Axler, and K. A. Ribet. Number theory: Volume I: Tools and diophantine equations. Vol. 560. Springer New York, 2007.
Stinson, Douglas R. Cryptography: Theory and Practice. 3rd ed., Chapman & Hall/CRC, 2006.
M. R. S., and M. J. Quadratic Residues and Their Applications in Cryptography. Journal of Number Theory, vol. 210, 2020, pp. 123-135.
Niven, Ivan, Herbert S. Zuckerman, and Hugh L. Montgomery. An introduction to the theory of numbers. John Wiley & Sons, 1991.
Rosen, Kenneth H.. Elementary number theory and its applications. United Kingdom, Addison-Wesley, 2000.
Childs, Lindsay N. A concrete introduction to higher algebra. New York: Springer, 2009.
Kraft, James, and Lawrence Washington. An introduction to number theory with cryptography. Chapman and Hall/CRC, 2018.
Gallian, Joseph A. Contemporary Abstract Algebra. Cengage Learning, 2016.
Hardy, G. H., and E. M. Wright. An Introduction to the Theory of Numbers. Oxford UP, 2008.
Lang, Serge. Algebra. Springer, 2002.
Silverman, Joseph H. A Friendly Introduction to Number Theory. Pearson, 2013.
Rezaei, M., et al. Quadratic residues graphs. International Journal of Pure and Applied Mathematics 113.3 (2017): 465-470.
Munkres, James R. Topology. 2nd ed., Pearson, 2018.
Willard, Stephen. General Topology. Dover Publications, 2012.
Engelking, Ryszard. General Topology. Revised and expanded edition, Heldermann Verlag, 2020.
Daoub, Hamza, Osama Shafah, and Fathi Brebish. A topological space defined on the group of units modulo p. unpublished, (2024).