# THE APPLICATIONS OF ZERO DIVISORS OF SOME FINITE RINGS OF MATRICES IN PROBABILITY AND GRAPH THEORY

## DOI:

https://doi.org/10.11113/jurnalteknologi.v83.14936## Keywords:

Zero divisor, ring theory, ring of matrices, graph theory, zero divisor graph## Abstract

Let *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.

## References

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.

## Downloads

## Published

## Issue

## Section

## License

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.

## How to Cite

*Jurnal Teknologi*,

*83*(1), 127-132. https://doi.org/10.11113/jurnalteknologi.v83.14936