The Zero Product Probability of Some Finite Ring of Matrices Based on the Order of the Annihilator
Abstract
An annihilator is defined as the set of pairs of elements in a ring R in which the product of the elements in the pair is the zero element of R. In this paper, we aim to determine the order of the annihilator of the finite ring of matrices of dimension two over integers modulo prime, M2(Zp). The order of the annihilator is determined by using number theory, specifically the linear congruence method. Another subject that we discuss in this paper is the zero product probability of a finite ring. The zero product probability is defined as the probability that two elements of a finite ring have product zero. Based on the order of the annihilator, the general formula of the zero product probability of M2(Zp) is determined.
Full text article
References
P. Erdos and P. Turan, “On some problem of a statistical group theory iv,” Acta Mathematica Academiae Scientiaruin Hungaricae, vol. 19, no. 3-4, pp. 413–435, 1968. https://doi.org/10.1007/BF00536750.
D. MacHale, “Commutativity in finite rings,” The American Mathematical Monthly, vol. 83, no. 1, pp. 30–32, 1976. https://doi.org/10.1080/00029890.1976.11994032.
S. M. Buckley and D. MacHale, “Commuting probability for subrings and quotient rings,” Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 4, no. 2, pp. 189-196, 2016. https://dergipark.org.tr/en/download/article-file/267255.
J. Dutta, D. K. Basnet, and R. K. Nath, “On commuting probability of finite rings,” Indagationes Mathematicae, vol. 28, pp. 372–382, 2017. https://doi.org/10.1016/j.indag.2016.10.002.
P. Dutta and R. K. Nath, “On relative commuting probability of finite rings,” Miskolc Mathematical Notes, vol. 20, no. 1, pp. 225–232, 2019. https://doi.org/10.48550/arXiv.1708.
P. Dutta and R. K. Nath, “A generalization of commuting probability of finite rings,” Asian-European Journal of Mathematics, vol. 11, no. 2, pp. 1850023(1–15), 2018. https://doi.org/10.1142/S1793557118500237.
S. Rehman, A. Q. Baig, and K. Haider, “A probabilistic approach toward finite commutative rings,” Southeast Asian Bulletin of Mathematics, vol. 43, p. 413–418, 2019.
S. M. S. Khasraw, “What is the probability that two elements of a finite ring have product zero?,” Malaysian Journal of Fundamental and Applied Sciences, vol. 16, no. 4, p. 497–499, 2020. https://doi.org/10.48550/arXiv.1810.13312.
H. M. Mohammed Salih, “On the probability of zero divisor elements in group rings,” International Journal of Group Theory, vol. 11, no. 4, p. 253–257, 2022. https://ijgt.ui.ac.ir/article_26054.html.
D. Dol˘zan, “The probability of zero multiplication in finite rings,” Bulletin of the Australian Mathematical Society, vol. 106, no. 1, p. 83–88, 2022. https://doi.org/10.1017/
S0004972721001246.
N. A. F. O. Zai, N. H. Sarmin, and N. Zaid, “The zero product probability of some finite rings,” Menemui Matematik, vol. 42, no. 2, p. 51–58, 2020. https://myjms.mohe.gov.my/index.php/dismath/article/download/13044/6742.
J. Fraleigh and V. Katz, A First Course in Abstract Algebra (7th Edition). Addison-Wesley, 2003.
N. H. McCoy, “Remarks on divisors of zero,” The American Mathematical Monthly, vol. 49, no. 5, p. 286–295, 1942. https://doi.org/10.1080/00029890.1942.11991226.
F. E. Hohn, Elementary Matrix Algebra. Collier-Macmillan, 1973.
N. Zaid, N. H. Sarmin, and S. M. S. Khasraw, “The zero product probability of ring of matrices based on euler’s phi-function,” Kongzhi yu Juece/Control and Decision, vol. 38, no. 7, pp. 2319–2326, 2023.
D. Burton, Elementary Number Theory. McGraw-Hill, 2002.
Authors
Copyright (c) 2025 Journal of the Indonesian Mathematical Society
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Article Details
