THE APPLICATIONS OF ZERO DIVISORS OF SOME FINITE RINGS OF MATRICES IN PROBABILITY AND GRAPH THEORY
Keywords:Zero divisor, ring theory, ring of matrices, graph theory, zero divisor graph
AbstractLet R be a finite ring. The zero divisors of R are defined as two nonzero elements of R, say x and y where xy = 0. Meanwhile, the probability that two random elements in a group commute is called the commutativity degree of the group. Some generalizations of this concept have been done on various groups, but not in rings. In this study, a variant of probability in rings which is the probability that two elements of a finite ring have product zero is determined for some ring of matrices over integers modulo n. The results are then applied into graph theory, specifically the zero divisor graph. This graph is defined as a graph where its vertices are zero divisors of R and two distinct vertices x and y are adjacent if and only if xy = 0. It is found that the zero divisor graph of R is a directed graph.
Hebisch, U. and Weinert, H. J. 1998. Semirings: Algebraic Theory and Applications in Computer Science. World Scientific Publishing: Singapore.
Hurley, B. and Hurley, T. 2011. Group Ring Cryptography. International Journal of Pure and Applied Mathematics. 69(1): 67-86.
Garces, Y., Torres, E., Pereira, O. and Rodriguez, R. 2014. Application of the Ring Theory in the Segmentation of Digital Images. Arxiv, arxiv.org/pdf/1402.4069.pdf
Fraleigh, J. B. and Katz, V. 2003. A First Course in Abstract Algebra (7th Edition). Addison-Wesley.
Matsumura, H. 1989. Commutative Ring Theory. University Press: Cambridge.
Gustafson, W. H. 1973. What is the Probability that Two Group Elements Commute. The American Mathematical Monthly. 80(9): 1031-1034.
Dutta, J., Basnet, D. K. and Nath, R. K. 2017. On Commuting Probability of Finite Rings. Indagationes Mathematicae. 28: 372-382.
Riaz, F. and Ali, K. M. 2011. Applications of Graph Theory in Computer Science. 2011 Third International Conference on Computational Intelligence, Communication Systems and Networks. 142-145.
Essam, J. W. 1971. Graph Theory and Statistical Physics. Discrete Mathematics. 1(1): 83-112.
Balaban, A. T. 1985. Applications of Graph Theory in Chemistry. Journal of Chemical Information and Modeling. 25(3): 334-343.
Sarmin, N. H., Mohd Noor, A. H. and Omer, S. M. S. 2017. On Graphs Associated to Conjugacy Classes of Some Three-Generator Groups. Jurnal Teknologi. 79(1): 55-61.
Andeson, D. F., Axtell, M. C. and Stickles, J. A. 2010. Zero-divisor Graphs in Commutative Rings. Commutative Algebra. 23 â€“ 45.
Rowen, L. H. 1988. Ring Theory (Vol. 1). Academic Press Inc: Boston.
Sherman, G. 1975. What is the Probability an Automorphism Fixes a Group Element. The American Mathematical Monthly. 82(3): 261-264.
Erfanian, A., Rezaei, R. and Lescot, P. 2007. On the Relative Commutativity Degree of a Subgroup of a Finite Group. Communications in Algebra. 35: 4183-4197.
Zamri, S. N. A., Sarmin, N. H., Khasraw, S. M. S., El-sanfaz, M. A. and Rahmat, H. 2017. On the Commutativity Degree of Metacyclic Groups of 5-Power Order Using Conjugation Action. Malaysian Journal of Fundamental and Applied Sciences. 13(4): 784-787.
MacHale, D. 1976. Commutativity in Finite Rings. The American Mathematical Monthly. 83(1): 30-32.
Buckley, S. M., MacHale, D. and Aine, N. S. 2014. Finite Rings with Many Commuting Pairs of Elements. Preprint.
Khasraw, S. M. S. 2018. What is the Probability that Two Elements of a Finite Ring have Product Zero. Preprint.
Bondy, J. and Murty, U. 1976. Graph Theory with Applications. Elsevier Science Publishing Co., Inc: New York.
Anderson, D. F. and Livingston, P. S. 1999. The Zero-Divisor Graph of a Commutative Ring. Journal of Algebra. 217: 434-447.
Redmond, S. P. 2002. The Zero-Divisor Graph of a Non-Commutative Ring. International Journal of Commutative Rings. 1(4): 203-211.
How to Cite
Copyright of articles that appear in Jurnal Teknologi belongs exclusively to Penerbit Universiti Teknologi Malaysia (Penerbit UTM Press). This copyright covers the rights to reproduce the article, including reprints, electronic reproductions, or any other reproductions of similar nature.